cdbook · §4.5. hopf algebra of chord diagrams 90 §4.6. hopf algebra of weight systems 95 §4.7....

CDBooKIntroduction to Vassiliev Knot Invariants

Not yet a draft: a raw text in statu nascendi.

Comments welcome. February 28, 2006

S. V. Chmutov

S. V. Duzhin

The Ohio State University, Mansfield Campus, 1680 Univer-

sity Drive, Mansfield, OH 44906

E-mail address: [email protected]

Steklov Institute of Mathematics, St. Petersburg Division,

Fontanka 27, St. Peterburg, 191011, Russia

E-mail address: [email protected]

Contents

Preface 8

Part 1. Fundamentals

Chapter 1. Knots and their relatives 15

§1.1. Definitions and examples 15

§1.2. Isotopy 16

§1.3. Plane knot diagrams 18

§1.4. Orientation 20

§1.5. Tangles and things 21

§1.6. Knot tables 25

§1.7. Algebra of knots 26

§1.8. Variations 28

Exercises 29

Chapter 2. Knot invariants 33

§2.1. Definition and first examples 33

§2.2. Linking number 34

§2.3. Alexander–Conway polynomial 37

§2.4. Jones polynomial 39

§2.5. Algebra of knot invariants 41

§2.6. Quantum invariants 43

§2.7. Two-variable link polynomials 49

Exercises 52

3

4 Contents

Chapter 3. Finite type invariants 55

§3.1. Discriminant approach 55

§3.2. Definition of Vassiliev invariants 59

§3.3. The Casson invariant 62

§3.4. Chord diagrams 64

§3.5. Goussarov’s approach 67

§3.6. Algebra of Vassiliev invariants 68

§3.7. Classical knot polynomials as Vassiliev invariants 69

§3.8. Actuality tables 74

§3.9. The case of framed knots TBW 76

Exercises 77

Chapter 4. Chord diagrams 81

§4.1. Four- and one-term relations 81

§4.2. The Fundamental Theorem 84

§4.3. Intersection graphs 86

§4.4. Bialgebras of knots and knot invariants 88

§4.5. Hopf algebra of chord diagrams 90

§4.6. Hopf algebra of weight systems 95

§4.7. Primitive elements in A 97

§4.8. Linear chord diagrams 98

Exercises 100

Part 2. Diagrammatic Algebras

Chapter 5. Jacobi diagrams 105

§5.1. Closed Jacobi diagrams 105

§5.2. IHX relation 108

§5.3. Isomorphism A ≃ C 112

§5.4. Product and coproduct in C 114

§5.5. Primitive subspace of C 115

§5.6. Open Jacobi diagrams 119

§5.7. Linear isomorphism B ≃ C 121

§5.8. Relation between A and B 127

§5.9. The three algebras in small degrees 129

Exercises 130

Chapter 6. Algebra of 3-graphs 133

Contents 5

§6.1. The space of 3-graphs 134

§6.2. Edge multiplication 135

§6.3. Vertex multiplication 141

§6.4. Action of Γ on the primitive space P 143

§6.5. Vogel’s algebra Λ 145

Exercises 149

Chapter 7. Lie algebra weight systems 151

§7.1. Lie algebra weight systems for the algebra A 151

§7.2. Lie algebra weight systems for the algebra C 162

§7.3. Lie algebra weight systems for the algebra B 170

§7.4. Lie algebra weight systems for the algebra Γ 174

§7.5. Weight systems not coming from Lie algebras 178

Exercises 180

Part 3. The Kontsevich Integral

Chapter 8. The Kontsevich integral 189

§8.1. First examples 189

§8.2. The construction 191

§8.3. Basic properties 193

§8.4. Example of calculation 194

§8.5. Convergence of the integral 196

§8.6. Invariance of the integral 197

§8.7. Horizontal multiplicativity 205

§8.8. Proof of the Kontsevich theorem 211

§8.9. Group-like property of the Kontsevich integral 215

Exercises 216

Chapter 9. Operations on knots and Kontsevich’s integral 221

§9.1. Reality of the integral 221

§9.2. Mirror reflection 223

§9.3. Orientations of knots 224

§9.4. Mutation 225

§9.5. Cablings 227

Exercises 234

Chapter 10. The Kontsevich integral: advanced topics 237

6 Contents

§10.1. Framed version of the Kontsevich integral 237

§10.2. Relation with quantum invariants 239

§10.3. The Kontsevich integral for braids TBW 239

§10.4. Canonical Vassiliev invariants 240

§10.5. Wheeling TBW 244

§10.6. The Kontsevich integral of the unknot 244

§10.7. Rozansky’s rationality conjecture 245

Exercises 246

Part 4. Other Topics

Chapter 11. The case of links 251

§11.1. Diagrammatic algebras: general case 251

§11.2. Relation to representation theory 252

§11.3. Vassiliev invariants of links 252

Exercises 252

Chapter 12. The Drinfeld associator 253

§12.1. The Knizhnik—Zamolodchikov equation and iteratedintegrals 253

§12.2. Calculation of the KZ Drinfeld associator 262

§12.3. Combinatorial construction of the Kontsevich integral 274

§12.4. General associators 283

Exercises 289

Chapter 13. Gauss diagrams 295

§13.1. Gauss diagrams and virtual knots 295

§13.2. The Jones polynomial via Gauss diagrams 297

§13.3. Arrow diagram formulas for Vassiliev invariants 298

§13.4. The algebra of arrow diagrams 302

Exercises 303

Chapter 14. Miscellany 305

§14.1. The Melvin–Morton conjecture 305

§14.2. The Goussarov–Habiro theory 313

§14.3. Bialgebra of intersection graphs 320

§14.4. Estimates for the number of Vassiliev knot invariants 323

Exercises 330

Contents 7

Appendix 333

§A.1. Lie algebras and their universal envelopes 333

§A.2. Bialgebras and Hopf algebras 336

§A.3. Free associative and free Lie algebras 348

Bibliography 351

Notations 361

Index 363

8 Contents

Preface

This text provides an introduction to the theory of finite type (Vassiliev)knot invariants, with a stress on its combinatorial aspects. It is intendedfor readers with no or little background in this area, and we care moreabout a clear explanation of the basic notions and constructions than aboutwidening the exposition to more recent and more advanced material. Ouraim is to lead the reader to understanding through pictures and calcula-tions rather than through explaining abstract theories. For this reason weallow ourselves to give the detailed proofs only if they are transparent andinstructive, referring more advanced readers to the original papers for moretechnical considerations.

Historical remarks. The notion of finite type knot invariants was inde-pendently invented by Victor Vassiliev (Moscow) and Mikhail Goussarov(St. Petersburg) around 1989–90 and first published in their respective pa-pers [Va1] (1990) and [G1] (1991).1

Neither Vassiliev nor Goussarov thought very much of their discovery.M. Goussarov (1958–1999), doing first-class works, somehow did not careabout putting his thoughts on paper. V. Vassiliev, at the time, was notmuch interested in knot invariants, either: his main concern was the generaltheory of discriminants in the spaces of smooth maps. It was V. I. Arnold[Ar2] who understood the revolutionary importance of finite type invariantsin knot theory. He coined the name “Vassiliev invariants” and told aboutthem Joan Birman, one of the world’s leading experts on knots. Since thattime, the term “Vassiliev invariants” became standard, although nowadayssome people use the expression “Vassiliev-Goussarov invariants”, which ismore close to the historical truth.

Vassiliev’s approach is based on the study of discriminants in the (infinite-dimensional) spaces of smooth maps from one manifold into another. Bydefinition, the discriminant consists of all maps with singularities.

For example, consider the space of all smooth maps from the circle into3-space M = f : S1 → R3. If f is an embedding (has no singularpoints), it represents a knot. The complement to the set of all knots is thediscriminant Σ ⊂ M. It consists of all smooth maps from S1 into R3 thathave singularities, either local, where f ′ = 0, or non-local, where f is notinjective. Two knots are equivalent if and only if they can be joined by apath in the space M that does not intersect the discriminant. Therefore,knot types are in one-to-one correspondence with the connected componentsof the complement M \ Σ, and knot invariants with values in an Abelian

1There is an important difference between the two variants of the theory — see Sec. 67 fordetails.

Preface 9

group G are nothing but cohomology classes fromH0(M\Σ,G). To computethis cohomology, V. Vassiliev makes a simplicial resolution of the singularvariety Σ, then introduces a filtration of this resolution by finite-dimensionalsubspaces that gives rise to a spectral sequence containing, in particular, thespaces of finite type invariants.

J. Birman and X.-S. Lin [BL] have contributed a lot to the simplificationof Vassiliev’s original techniques. They explained the relation between Jonespolynomial and finite type invariants and emphasized the role of the algebraof chord diagrams. M. Kontsevich [Kon1] introduced an analytical tool(Kontsevich’s integral) to prove that the study of Vassiliev invariants can bereduced to the combinatorics of chord diagrams.

D. Bar-Natan was the first mathematicians who undertook a thoroughstudy of Vassiliev invariants as such. In his preprint [BN0] and PhD thesis[BNt] he found the relation between finite type invariants and topologicalquantum field theory elaborated by his scientific advisor E. Witten [Wit1,Wit2]. Bar-Natan’s paper [BN1] (whose preprint edition [BN1a] appearedin 1992) is still the most authoritative source on the fundamentals of thetheory of Vassiliev invariants. About the same time, T. Le and J. Murakami[LM2], relying on V. Drinfeld’s work [Dr1, Dr2], proved the rationality ofthe Kontsevich integral.

Among more recent important developments in the area of finite typeknot invariants let us mention:

• The existence of non-Lie-algebraic weight systems (P. Vogel [Vo],J. Lieberum [Lieb]).• J. Kneissler’s fine analysis [Kn1, Kn2, Kn3] of the structure of

the algebra Λ introduced by P. Vogel [Vo].• Gauss diagram formulas invented by M. Polyak and O. Viro [PV1]

and the proof by M. Goussarov [G3] that all finite type invariantscan be obtained in this way.• D. Bar-Natan’s proof that Vassiliev invariants for braids separate

braids [BN4]2 (this is the only situation where Vassiliev invariantsare proved to be a complete systems of invariants).• Habiro’s theory of claspers [Ha2] (see also [G4]).• V. Vassiliev’s papers [Va4, Va5] where a general technique for de-

riving combinatorial formulas for cohomology classes in the comple-ments to discriminants, and in particular, for finite type invariants,is proposed.• Explicit formulas for the Kontsevich integral of some knots and

links ([BGRT, BLT, BNL, Roz2, Kri2, Mar, GK].

2This fact is also an immediate consequence of T. Kohno’s theorem [Koh2], proved beforethe notion of finite type invariants was introduced.

10 Contents

An important source of regularly updated information on finite type in-variants is the online Bibliography of Vassiliev invariants started by D. Bar-Natan and currently living at

http://www.pdmi.ras.ru/~duzhin/VasBib/

On July 12, 2005 it contained 600 entries, out of which 36 were added duringthe seven months of 2005. The study of finite type invariants is going on at asteady pace. However, notwithstanding all efforts two of the most importantproblems risen in 1990 are still open:

(1) Is it true that Vassiliev invariants distinguish knots?

(2) Is it true that Vassiliev invariants can detect knot orientation?

Prerequisites. We assume that the reader has basic knowledge of calcu-lus on manifolds (vector fields, differential forms, Stokes’ theorem), generalalgebra (groups, rings, modules, Lie algebras, fundamentals of homologi-cal algebra), linear algebra (vector spaces, linear operators, tensor algebra,elementary facts about representations) and topology (topological spaces,homotopy, homology, Euler characteristic). More advanced algebric mater-ial (bialgebras, free algebras, universal enveloping algebras etc.) which is ofprimary importance in this book, can be found in the Appendix at the endof the book. No knowledge of knot theory is presupposed, although it maybe useful.

Contents. The book consists of four parts, divided into fourteen chapters.

The first part opens with a short introduction into the theory of knotsand their classical polynomial invariants and closes with the definition ofVassiliev invariants.

In part 2, we systematically study the graded Hopf algebra naturallyassociated with the filtered space of Vassiliev invariants, which appears inthree different disguises: as the algebra of chord diagrams A, as the algebraof closed diagrams C, and as the algebra of open 1-3-diagrams B. After that,we study the auxiliary algebra Γ generated by regular trivalent graphs andclosely related to the algebras A, B, C as well as to Vogel’s algebra Λ. Inthe last chapter we discuss the weight systems defined by Lie algebras, bothuniversal and depending on a chosen representation.

Part 3 is dedicated to a detailed exposition of the Kontsevich integral;it contains the proof of the main theorem of the theory of Vassiliev knotinvariants that reduces their study to combinatorics of chord diagrams andrelated algebras. In Chapter 10 we discuss more advanced material relatedto the Kontsevich integral: the wheels formula, the Rozansky conjecture etc.

The last part of the book is devoted to various topics left out in the previ-ous parts, in particular, Gauss diagram formulas for Vassiliev invariants, the

Preface 11

Melvin-Morton conjecture, the Drinfeld associator, the Goussarov–Habirotheory, the size of the space of Vassiliev invariants etc.

The book is intended to be a textbook, so we have included many exer-cises, both embedded in text and constituting a separate section at the endof each chapter. Open problems are marked with an asterisk.

Chapter dependence.

NA

IB

MO

C

TO

RC

IS

ORTN

U C T ID

IO

N

AT

YC

SI

LN

A

Ch.11 The case of links

Ch.2

Gauss diagramsCh.13 MiscellanyCh.14

Ch.8 The Kontsevich integral

Ch.7 Lie algebraweight systems

Ch.6 Algebra of3-graphs

Ch.5diagrams

Jacobi

Ch.3

Ch.4

Operations on knotsand Kontsevich's integral

Ch.10

LAO T O P IN

C SITIDDA

Ch.1

Ch.12 The Drinfeld associator

Knots and relatives

Finite type invariants

Knot invariants

Chord diagrams

Ch.9

Kontsevich integral:advanced topics

12 Contents

If this text is used as an actual textbook, then a one semester coursemay consist of the introduction (chapters 1–4) followed by either combina-torial (5–7) or analytical (8–10) chapters according to the preferences of theinstructor and the students. If time permits the course may be enhancedby some of the additional topics. For a one year course we would recom-mend to follow the numerical order of chapters 1–9 and conclude with someadditional topics to teacher’s taste.

Acknowledgements. Our work on this book actually began in August1992, when our colleague Inna Scherbak returned to Pereslavl-Zalessky fromthe First European Mathematical Congress in Paris and brought a photo-copy of Arnold’s lecture notes about the new-born theory of Vassiliev knotinvariants. We spent several months filling our waste-paper baskets withpictures of chord diagrams, before our first joint article [CD1] was ready.

In the preparation of the present text, we have extensively used our pa-pers (joint, single-authored and with other coauthors, see bibliography) andin particular, lecture notes of the course “Vassiliev invariants and combina-torial structures” that one of us (S. D.) delivered at the Graduate Schoolof Mathematics, University of Tokyo, in Spring 1999. It is our pleasureto thank V. I. Arnold, D. Bar-Natan, J. Birman, C. De Concini, O. Das-bach, V. Goryunov, T. Kerler, T. Kohno, S. Lando, M. Polyak, I. Scherbak,A. Vaintrob, A. Varchenko, V. Vassiliev and S. Willerton for many usefulcomments concerning the subjects touched upon in the book. We are like-wise indebted to the anonymous referees whose criticism and suggestionshelped us to improve the text. Our work was supported by several grants:INTAS 00-0259, NWO-RFBR 047.008.005, RFBR 00-15-96084 and 01-01-00660 (S. D.), Professional Development Grant of OSU, Mansfield (S. Ch.).Part of the work was accomplished when the second author was visiting theOhio State University (autumn quarter 2003) and Max-Planck-Institut furMathematik (summer 2005).

Part 1

Fundamentals

Chapter 1

Knots and their

relatives

This book is about knots. It is, however, hardly possible to speak aboutknots without mentioning other one-dimensional topological objects embed-ded in space. Therefore, in this introductory chapter we give basic definitionsand constructions pertaining to knots and their relatives: links, braids andtangles.

The table of knots presented at the end of this chapter (page 27) will beused throughout the book as the source of examples and exercises.

1.1. Definitions and examples

A knot is a closed non-self-intersecting curve in 3-space. In this book, wewill mainly study smooth oriented knots, and an exact definition can begiven as follows.

1.1.1. Definition. A knot is an embedding f : S1 → R3, where S1 is anoriented circle and R3 is oriented space.

Recall that an embedding is a smooth map which is injective and whosederivative (in our case, tangent vector to the curve) is nowhere zero.

Forgetting the orientations, we arrive at the notion of a non-orientedknot in non-oriented space. One can also consider oriented knots in non-oriented space and non-oriented knots in oriented space. Note that thereis no difference between these notions viewed as set-theoretic objects; whatis different is the equivalence between knots which may or may not depend

15

16 1. Knots and their relatives

on the orientations. We will discuss various equivalence relations betweenknots in the next section.

In the definition above and everywhere in the sequel, the word smoothmeans infinitely differentiable.

If coordinates are chosen in R3, a knot can be given by 3 smooth periodicfunctions of one variable x(t), y(t), z(t).

1.1.2. Example. The simplest knot is represented by a plane circle:

x = cos t,y = sin t,z = 0.

x

yz

1.1.3. Example. The curve that goes 3 times around and 2 times across astandard torus in R3 is called the trefoil knot, or the (2, 3)-torus knot:

x = (2 + cos 3t) cos 2t,y = (2 + cos 3t) sin 2t,z = sin 3t.

1.1.4. Exercise. Give the definition of a (p, q)-torus knot. For what valuesof p and q does this definition make sense?

Unless otherwise specified, all knots are assumed to be oriented: thepositive direction is that which corresponds to the counterclockwise motionaround the parameterizing circle S1 = (cos t, sin t)|t ∈ R ⊂ R2.

1.2. Isotopy

A smooth parametrized curve, considered by itself, falls within the scope ofdifferential geometry. The topological study of knots presupposes that theyshould be considered up to an equivalence relation which (1) discards thespecific choice of parameterization, (2) models the physical transformationsof a closed piece of rope in space.

1.2.1. Definition. A smooth isotopy of a knot K, given by a mappingf : S1 → R3, is a smooth family of knots fu, u ∈ R, such that f0 = f .

By a smooth family of maps, or a map smoothly depending on a para-meter, we understand a smooth mapping F : S1 × I → R3, where I ⊂ R isa real interval. Assigning a fixed value u to the second argument of F , weget a mapping fu : S1 → R3.

1.2. Isotopy 17

For example, equations

(1)x = (u+ cos(3t)) cos(2t),y = (u+ cos(3t)) sin(2t),z = sin(3t),

where u ∈ (1,+∞), represent a smooth isotopy of the above consideredtrefoil knot, which corresponds to u = 2:

u = 2 u = 1.5 u = 1.2 u = 1

For any u > 1 we have a knot (smooth closed curve without self-intersections), but as soon as the value u = 1 is reached we get a singularcurve with three coinciding cusps corresponding to the values t = π/3, t = πand t = 5π/3. This curve is not a knot.

1.2.2. Definition. Two knots are said to be isotopic if one can be trans-formed into another by means of a smooth isotopy.

Example. This picture shows an isotopy of the figure eight knot into itsmirror image:

1.2.3. There are other notions of knot equivalence, namely, ambient equiv-alence and ambient isotopy, which, for smooth knots, are the same thing asequivalence defined through isotopy. Here are the corresponding definitions.Proofs can be found in [BZ].

Definition. Two knots, f and g, are ambient equivalent if there is a com-mutative diagram

S1 f−−−−→ R3

ϕ

yyψ

S1 g−−−−→ R3

where ϕ and ψ are orientation preserving diffeomorphisms of the circle andspace, respectively.

Definition. Two knots, f and g, are ambient isotopic if there is a smoothfamily of diffeomorphisms of 3-space Ht : R3 → R3 such that H0 = id andH1 f = g.


1.2.4. A knot equivalent to the plane circle of Example 1.1.2 is referred toas a trivial knot, or unknot.

Looking at a picture of a trivial knot, it is sometimes not easy to under-stand that the knot is indeed trivial:

Trivial knots

There are algorithmic procedures to detect whether a given knot diagramrepresents the unknot; one of them is, based on W. Thurston’s ideas, isimplemented in J. Weeks’ computer program SnapPea, see [Wee].

Here are some more examples of knots.

Left trefoil Right trefoil Figure 8 knot Granny knot Square knot

1.2.5. Exercise. Is the torus knot described in section 1.1.3, a right or aleft trefoil?

Knots are a special case of links.

1.2.6. Definition. A link is a smooth embedding S1⊔· · ·⊔S1 → R3, whereS1 ⊔ · · · ⊔ S1 is the disjoint union of several circles.

Trivial 2-component link Hopf link Whitehead link Borromean link

Equivalence of links is defined in the same way as for knots — with theexception that now one may, or may not, distinguish between the compo-nents of a link and thus speak about the equivalence of links with numberedor unnumbered components.

In the future, we will often say ‘knot’ (‘link’) instead of ‘equivalenceclass’, or ‘topological type of knots (links)’.

1.3. Plane knot diagrams

To visualize knots and links on paper, we use knot diagrams which areformally defined as plane projections with only transversal double points(crossings) as singularities, and at each crossing one bit of information is

1.3. Plane knot diagrams 19

added to tell which branch is higher (overpass) and which is lower (under-pass).

1.3.1. Theorem (Reidemeister [Rei]). In terms of knot diagrams, two non-oriented knots, K1 and K2, are equivalent if and only if a diagram of K1 canbe transformed into a diagram of K2 by a sequence of plane diffeomorphismsand local moves of the following three types:

Ω1 Ω2 Ω3

Reidemeister moves

To adjust the assertion of this theorem to the oriented case, each ofthe three Reidemeister moves has to be equipped with orientations in allpossible ways. See [Oll] for details.

Exercise. Find an isotopy and determine the sequence of Reidemeistermoves that relates two different diagrams of the trefoil knot:

1.3.2. Local writhe. Crossing points on a diagram come in two species,positive and negative, as illustrated in the picture:

Positive crossing Negative crossing

Although this sign is defined in terms of the knot orientation, it is easyto check that it does not change if the orientation is reversed. For links withmore than one component, the choice of oientation is essental.

The local writhe of a crossing is defined as the number equal to +1 or−1 for positive or negative points, respectively. The writhe (or total writhe)of a diagram is the sum of the writhes of all crossing points, equal to thedifference between the number of positive and negative crossings. Of course,different diagrams of one and the same knot may have different total writhes.In the sequel (Chapter 2) we will use the writhe to construct knot invariants.


1.3.3. Alternating knots. A knot diagram is called alternating if its over-crossings and undercrossings alternate while we are traveling along the knot.A knot is called alternating if it has an alternating diagram. A knot diagramis called reducible there is a simple closed curve in the plane that intersectsthe knot diagram in exactly two points lying on two different branches neara crossing. The number of crossings in a reducible diagram can be decreasedby a move shown in the picture:

simple closed curve

reducible diagram reduction

A diagram which is not reducible is called reduced. As there is no imme-diate way to simplify a reduced diagram, the following conjecture naturallyarises (P. G. Tait, 1898).

The Tait conjecture. A reduced alternating diagram has the minimalnumber of crossings among all diagrams of the given knot.

This conjecture stood open for almost 100 years. It was proved only in1986 using the newly invented Jones polynomials simultaneously by threemathematicians: L. Kauffman [Ka6], K. Murasugi [Mur], and M. Thistleth-waite [Th] (see Exercise (12) in Chapter 2).

1.4. Orientation

As stated before, we are studying knots equipped with two orientations:that of the parameterizing circle and that of the ambient space. Changingone or another is an operation on knots that might or might not change theequivalence class of the knot.

Denote by K∗ = τ(K) the inverse of the knot K and by K = σ(K) themirror reflection of K. The operators τ and σ are involutions on the set of(equivalence classes of) knots; they generate a group of order four consistingof id, τ, σ and τ σ and isomorphic to ∼= Z2 ⊕Z2); the symmetry propertiesof a knot depend on its orbit under the action of this group.

1.4.1. Definition. A knot is called:

• invertible if it does not change when the direction of motion alongthe knot is reversed: K∗ = K,

• plus-amphicheiral, if K = K,

1.5. Tangles and things 21

• minus-amphicheiral, if K = K∗,

• fully symmetric, if K = K∗ = K = K∗,

• asymmetric, if all knots K, K∗, K, K∗

are different.

Example. The trefoil knots (both left and right) are invertible, becausethe rotation through 180 around an axis in R3 changes the direction of thearrow on the knot.

The existence of non-invertible knots was first proved by H. Trotter [Tro]in 1964. Among the knots with up to 8 crossings (see Table 1 on page 27)there is only one non-invertible knot: 817; this fact was proved in 1979 byA. Kawauchi [Ka1].

Example. The trefoil knot is not amphicheiral (this is why we distinguishbetween the left and the right trefoil). A proof of this fact, based on thecalculation of Jones’ polynomial, will be given in Sec. 2.4.

Example. The figure eight knot is amphicheiral. This is proved by theisotopy shown in the picture on page 17.

Among the 35 knots with up to 8 crossings shown in Table 1 there areexactly 7 amphicheiral knots: 41, 63, 83, 89, 812, 817, 818, out of which 817

is minus-amphicheiral, the rest, as they are invertible, are both plus- andminus-amphicheiral.

The simplest totally asymmetric knots appear in 9 crossings, they are932 and 933. The following are all non-equivalent:

933 9∗33 933 9∗33

1.5. Tangles and things

A tangle is a notion important in knot theory for two reasons: first, it is ageneralization of a knot, secondly, knots can be represented as combinationsof (simple) tangles.

1.5.1. Definition. A tangle is a one-dimensional compact piecewise smoothsubmanifold of the slice C × [a, b] ⊂ R3 whose boundary points belong tothe boundary of the slice, considered up to an isotopy of the slice, identicalon the boundary, and homotheties of the slice along the interval [a, b]. Weassume that the tangent line of the tangle may be horizontal at only a finite


number of points, and this can never happen at the boundary points (thetop and the bottom),

1.5.2. Operations. In the case when the top of one tangle T1 coincideswith the bottom of another tangle T2 (for oriented tangles we require thecoincidence of orientations, too), one can define the product T1 ·T2 by puttingT1 on top of T2:

T1 = ; T2 = ; T1 ·T2 = .

Another operation, tensor product , is defined as placing one tangle nextto the other:

T1 ⊗ T2 = .

1.5.3. Special types of tangles. Knots, links and braids are particularcases of tangles. A link is the same thing as a tangle without boundary; ifit is, moreover, connected, then we get a knot.

Definition. A braid on n strings is a tangle consisting of n components,each with one boundary point on the bottom and one boundary point on thetop with extra requirements that (1) each string has a one-to-one projectiononto the segment [0, 1] (the axis of the slice). (2) the boundary points occupycertain fixed positions in the horizontal planes, so that braids with the samenumber of strings can always be multiplied together.

One more notion occupying an intermediate position between braids andgeneral tangles is that of a string link. This is a tangle consisting of severaltopological intervals, each connecting a point on the top with a point onthe bottom. The distinction with braids is the components may be knotted.Here is a formal definition.

Definition. Let D2 be a disk in R2 and p1, ..., pn ∈ D2. An n componentstring link is an embedding of the union of n intervals into the productD2 × [0, 1] such that the endpoints of the i-th interval go into pi × 0 andpi × 1, considered up to an isotopy, fixed on the boundary.

A comparison chart:

A link A knotA braidA tangle A string link

1.5. Tangles and things 23

1.5.4. Braids. Braids are useful in the study of knots, because any knotcan be represented as a closure of a braid (Alexander’s theorem [Al1]):

Braids are in many respects easier to work with, as they form groupsunder tangle multiplication: the set of equivalence classes of braids on nstrings is the braid group denoted by Bn. The group Bn is generated by theelements σi, i = 1, . . . , n− 1:

...

1i

...

i+

with the following complete set of relations.

Far commutativity, σiσj = σjσi, for |i− j| > 1.

Braiding relation, σiσi+1σi = σi+1σiσi+1, for i = 1, 2, . . . , n− 2.

Number the bottom endpoints and the top endpoints of the braid by1, . . . , n from left to right. Then any braid b ∈ Bn yields a permutationσ of 1, . . . , n: take the string number i at the bottom, then the numberof its top point is σ(i). For example, the braid σ1σ2 sends 1 to 2, 2 to 3,and 3 to 1. This gives an epimorphism π : Bn → Sn of the braid group ontothe group Sn of all permutations of 1, . . . , n. A braid is called pure if itbelongs to the kernel of the homomorphism π.

Theorem (Markov [Mark, Bir1]). Two closed braids are equivalent (aslinks) if and only if the braids are related by a finite sequence of the followingMarkov moves.

(M1) b←→ aba−1 for any a, b ∈ Bn;

(M2) Bn ∋ ...b ←→

...b ∈ Bn+1 ,

...b ←→

...b .


1.5.5. Elementary tangles. A link can be cut into several simple tanglesby a finite set of horizontal planes, and the link is equal to the product ofall such tangles. Every simple tangle is a tensor product of the followingelementary tangles.

Unoriented case:

id := , X+ := , X− := , max := , min := .

Oriented case:

id := , id∗ := , X+ := , X− := ,

−→max := ,

←−max := , min−→ := , min←− := .

For example, the generator σi ∈ Bn of the braid group is a simple tanglerepresented as the tensor product, σi = id⊗(i−1) ⊗X+ ⊗ id⊗(n−i−1).

1.5.6. The Turaev moves. Having a presentation of a tangle as a prod-uct of simple tangles it is natural to ask for an analog of Reidemeister’s(1.3.1) and Markov’s (1.5.4) theorems, i.e. when two such presentationsgive isotopic tangles. Here is the answer.

Theorem ([Tur3, FrYe]). Two products of simple tangles are isotopic ifand only if they are related by a finite sequence of the following Turaevmoves.

Turaev moves (unoriented case):

(T0)...

...

T1

T2

←→...

...

T1

T2

(T1) ←→ ←→ (id⊗max)·(X+⊗id)·(id⊗min)=id=

=(id⊗max)·(X−⊗id)·(id⊗min)

(T2) ←→ ←→ X+·X−=id⊗id=X−·X+

(T3) ←→ (X+⊗id)·(id⊗X+)·(X+⊗id)=(id⊗X+)·(X+⊗id)·(id⊗X+)

(T4) ←→ ←→ (max⊗id)·(id⊗min)=id=(id⊗max)·(min⊗id)

(T5) ←→ (id⊗max)·(X+⊗id)=(max⊗id)·(id⊗X−)

(T5′) ←→ (id⊗max)·(X−⊗id)=(max⊗id)·(id⊗X+)

1.6. Knot tables 25

Turaev moves (oriented case):

(T0) Same as in the unoriented case with arbitrary orientations of par-ticipating strings.

(T1— T3) Same as in the unoriented case with orientations of all stringsfrom bottom to top.

(T4) ←→ ←→ (−→max⊗id)·(id⊗min−→)=id=(id⊗←−max)·(min←−⊗id)

(T4′) ←→ ←→ (←−max⊗id∗)·(id∗⊗min←−)=id∗=(id∗⊗−→max)·(min−→⊗id∗)

(T5) ←→ (←−max⊗id⊗id∗)·(id∗⊗X+⊗id∗)·(id∗⊗id⊗min←−)··(id∗⊗id⊗−→max)·(id∗⊗X−⊗id∗)·(min−→⊗id⊗id∗)=id⊗id∗

(T6) ←→

(←−max⊗id∗⊗id∗)·(id∗⊗←−max⊗id⊗id∗⊗id∗)··(id∗⊗id∗⊗X+⊗id∗⊗id∗)··(id∗⊗id∗⊗id⊗min←−⊗id∗)·(id∗⊗id∗⊗min←−) =

= (id∗⊗id∗⊗−→max)·(id∗⊗id∗⊗id⊗−→max⊗id∗)··(id∗⊗id∗⊗X+⊗id∗⊗id∗)··(id∗⊗min−→⊗id⊗id∗⊗id∗)·(min−→⊗id∗⊗id∗)

1.6. Knot tables

1.6.1. Connected sum. There is a natural way to construct a ‘compli-cated’ knot from two ‘simple’ ones: each of the two knots is cut at somepoint, then the two pairs of loose ends are connected. This must be donewith some caution: first, by a smooth isotopy, both knots are deformed sothat for a certain regular plane projection they look as shown in the picturebelow on the left, then they are changed inside the dashed disk as shown onthe right:

The connected sum makes sense only for oriented knots. It is well-definedand commutative on the equivalence classes of knots.

Example. Taking the sum of two trefoils one can obtain two differentnon-oriented knots (or three different oriented knots):


#31 31#Granny knot, Square knot, 31 31

1.6.2. Definition. A knot is called prime if it cannot be represented as theconnected sum of two nontrivial knots.

Any knot is a connected sum of prime knots, and this decomposition isunique (see, e.g., [CrF] for a proof). In particular, this means that a trivialknot cannot be decomposed into two nontrivial knots. Therefore, to knowall knots, it is enough to have a table of all prime knots.

Conventionally, prime knots are tabulated as follows. First, all knotsthat have a plane diagram with 3 crossings are listed, then all knots withthe minimum of 4 crossings etc. Within each group, knots are numbered insome way. In Table 1, we use the so called Rolfsen numbering established in[Rol]) (in fact, this numbering goes back to the table compiled by Alexanderand Briggs in 1927). We also follow Rolfsen’s conventions in the choice ofthe version of non-amphicheiral knots: for example, our 31 is the left, notthe right, trefoil.

Rolfsen’s table of knots, authoritative as it is, still contained one error. Itis the famous Perko pair (knots 10161 and 10162 in Rolfsen) — two equivalentknots that were thought to be different during 75 years, beginning from 1899:

The equivalence of these two knots was established in 1973 by K. A. Perko[Per1], a lawyer from New York who was an undergraduate student of math-ematics at Princeton in 1960–1964 [Per2] but later chose jurisprudence tobe his profession. The combination of a professional lawyer and an amateurmathematician in one person is not new in the history of mathematics (thinkof Pierre Fermat!) and often leads to unexpected discoveries.

1.7. Algebra of knots

Denote by K the set of equivalence classes of knots. It forms a commutativesemigroup under the connected sum of knots. Given a field (or a commuta-tive ring) F, we can construct the semigroup algebra X = F[K] of K over F.By definition, elements of X are formal linear combinations

∑λiKi, λi ∈ F,

1.7. Algebra of knots 27

31 41(a) 51 52 61 62 63(a)

71 72 73 74 75 76 77

81 82 83(a) 84 85 86 87

88 89(a) 810 811 812(a) 813 814

815 816 817(an) 818(a) 819 820 821

Table 1. Non-oriented prime knots with up to 8 crossings. Am-phicheiral knots are marked by ‘a’, non-invertible knots by ‘n’.

Ki ∈ K, the product is defined by (K1,K2) 7→ K1#K2 on the basic vectorsand then extended by linearity to the entire X . This algebra X will bereferred to as the algebra of knots.

The algebra of knots provides a convenient language for the study of knotinvariants (see next Chapter): in these terms, a knot invariant is nothingbut a linear functional on X , i.e. an element of the dual space X ∗. Algebrahomomorphisms X → C, i.e.invariants that also respect multiplication, arereferred to as multiplicative invariants; later, in Section 4.4,we will see theimportance of this notion.


In the sequel, we will introduce more operations in this algebra as wellas in the dual algebra of knot invariants. We will also study a filtration inX that is closely related to the notion of a finite type knot invariant.

1.8. Variations

1.8.1. Framed knots. A framed knot is a knot equipped with a framing,i.e. a smooth section of its normal bundle, considered up to smooth defor-mations. One way to choose such a section is to use the blackboard framing,defined by a plane knot projection, with the framing always parallel to theprojection plane, e.g.

A framed knot can also be vizualised as a ribbon knot, i.e. a narrowknotted strip (see the right picture above).

An arbitrary framed knot can be drawn by a plane diagram with theblackboard framing. This is achieved by choosing an arbitrary projectionand then performing local moves to straighten out the twisted band:

,

For framed knots (with blackboard framing) the Reidemeister theorem 1.3.1is not true because the first Reidemeister move Ω1 changes the blackboardframing. Here is an appropriate substitute.

1.8.2. Theorem (framed Reidemeister theorem). Two knot dia-grams with blackboard framing D1 and D2 are equivalent if and only if D1

can be transformed into D2 by a sequence of plane isotopies and local movesof three types FΩ1, Ω2, and Ω3, where

FΩ1 :

One may also consider tangle diagrams with blackboard framing. Forsuch tangles there is an analog of Theorem 1.5.6 — the Turaev move (T1)should be replaced by its framed version that mimics the move FΩ1.

Exercises 29

1.8.3. Long knots. A long knot is a smooth embedding γ of the line R1

into R3 which coincides with a standard linear embedding near infinity:γ(x) = (x, 0, 0) for |x| > C. Two long knots are said to be equivalent, ifthere is a smooth isotopy of space which preserves the line X = (x, 0, 0)for big values of x and takes one knot into another. Instead of long knots,one can study embeddings of an interval into space with both endpointsfixed; in this sense long knots can be viewed as one-component string links.

A long knot can be turned into a conventional (closed) knot by replacinga neighbourhood of infinity on the line X by an arc of a sufficiently bigcircle. It is easy to prove that this map provides a one-to-one correspondencebetween the sets of equivalence classes of long and closed knots, thereforethe two theories are isomorphic.

Many constructions are easier to perform in the case of long knots. Forexample, the cut and paste operation for the connected sum becomes asimple concatenation.

1.8.4. Knots in arbitrary manifolds. This book is about knots in 3-space R3. Instead of R3 one can, with an equal success, study knots inthe 3-sphere S3, because the 1-point compactification R3 → S3 establishesa one-to-one correspondence between equivalence classes of knots in bothmanifolds.

If M is an arbitrary 3-manifold, then there exists a theory of knots inM which is different from the conventional case, see, e.g., [Kal, Va6].

If the dimension of the manifold M is different from 3, then the theoryof knots proper in M is trivial: any embedding S1 → M is isotopic to oneof the standard embeddings that represent the elements of the fundamentalgroup π1(M). The real analog of knot theory for 2-manifolds is Arnold’stheory of immersed curves [Ar3]; for higher dimensional manifolds one canstudy multidimensional knots, like embeddings S2 → R4, see, e.g. [Rol].

Exercises

(1) Find the following knots in table 1:

(a) (b) (c)

(2) Can you find the following links in the picture on page 18?


(3) Table 1 shows 35 topological types of non-oriented knots in non-orientedspace. What is the corresponding number of types of oriented knots inoriented space?

(4) Find an isotopy that transforms the knot 63 into its mirror image 63.

(5) Repeat Perko’s achievement: find an isotopy that transforms one of theknots of the Perko pair into another one.

(6) Decompose the knot on the right into a connectedsum of prime knots from Table 1.

(7) Show that by changing some crossings from overpass to underpass orvice versa, any knot diagram can be made into a diagram of the unknot.

(8) (C. Adams [AdC]) Show that by changing some crossings from overpassto underpass or vice versa, any knot diagram can be made alternating.

(9) Represent the knots 41, 51, 52 as closed braids.

(10) Find a sequence of Markov moves that transforms the closure of thebraid σ2

1σ32σ

41σ2 into the closure of the braid σ2

1σ2σ41σ

32.

(11) Garside’s fundamental braid ∆ ∈ Bn is defined as∆ := (σ1σ2 . . . σn−1)(σ1σ2 . . . σn−2) . . . (σ1σ2)(σ1) .

∆ =

(a) Prove that σi∆ = ∆σn−i for every standard generator σi ∈ Bn.(b) Prove that ∆2 = (σ1σ2 . . . σn−1)

n.(c) Check that ∆2 belongs to the center Z(Bn) of the braid group.(d) Show that any braid can be represented as a product of a certain

power (possibly negative) of ∆ and a positive braid, that containsonly positive powers of standard generators σi.

In fact, for n ≥ 3, the center Z(Bn) is an infinite cyclic group generatedby ∆2. The word and conjugacy problems in the braid group were solvedby F. Garside [Gar]. The structure of positive braids that occur in thelast statement was studied in [Adya, ElMo].

(12) (a) We denoted by π the natural epimorphism of the braid group Bnonto the permutation group Sn. Prove that the sign of the permu-tation π(b) is equal to the parity of the number of crossings of the

braid b, that is (−1)ℓ(b), where ℓ(b) is the length of the braid b as aword in generators σ1, . . . , σn−1.

(b) Prove that the subgroup of pure braids is gener-ated by the braids Aij , where the i-th and j-thstrings are linked with each other behind all otherstrings.

Aij =

i j

i

...

j

......

Exercises 31

(13) With each elementary generator σi ∈ Bn associate a linear transforma-tion Ξi of a vector space V of dimension n with a distinguished basise1, . . . , en which acts as a counterclockwise 90 rotation in the plane〈ei, ei+1〉: Ξi(ei) = ei+1, Ξi(ei+1) = −ei, Ξi(ej) = ej for j 6= i, i + 1.Prove that this construction gives a representation Bn → GLn(R) ofthe braid group.

(14) Consider a free module over the ring of Laurent polynomials Z[t±1] witha basis e1, . . . , en. The Burau representation Bn → GLn(Z[t±1]) sendsσi ∈ Bn into a linear operator that transforms ei to (1− t)ei+ ei+1, andei+1 to tei.(a) Prove that it is indeed a representation of the braid group.(b) The Burau representation is reducible. Its splits into the trivial

one-dimensional representation and (n−1)-dimensional irreduciblerepresentation which is called reduced Burau representation.Find a basis of the reduced Burau representation where the matriceshave the form

σ1 7→(−t t ... 0

0 1 ... 0....... . .

...0 0 ... 1

), σi 7→

1. . .1 0 01 −t t0 0 1. . .

1

, σn−1 7→

(1 ... 0 0...

. . ....

...0 ... 1 00 ... 1 −t

)

Answer. te1 − e2, te2 − e3, . . . , ten−1 − enThe Burau representation is faithful for n 6 3 [Bir1], and not faithfulfor n > 5 [Big1]. The case n = 4 remains open.

(15) Let V be a free Z[q±1, t±1] module of dimension n(n− 1)/2 with a basisei,j for 1 6 i < j 6 n. The Krammer–Bigelow representation can bedefined via the action of σk ∈ Bn on V :

σk(ei,j) =

8>>>>>>>>>>><>>>>>>>>>>>:ei,j if k < i− 1 or k > j,

ei−1,j + (1− q)ei,j if k = i− 1,

tq(q − 1)ei,i+1 + qei+1,j if k = i < j − 1,

tq2ei,j if k = i = j − 1,

ei,j + tqk−i(q − 1)2ek,k+1 if i < k < j − 1,

ei,j−1 + tqj−i(q − 1)ej−1,j if i < k = j − 1,

(1− q)ei,j + qei,j+1 if k = i− 1.

Prove that this assignment determines a representation of the braidgroup. It was shown in [Big2, Kram] that this representation is faithfulfor any n > 1. Therefore the braid group is a linear group.

(16) Represent the knots 41, 51, 52 as products of simple tangles.

(17) Decompose the tangle into elementary tangles.

(18) Consider the following two knots given as products of simple tangles:

(←−max⊗−→max )·(id∗⊗X+⊗id∗)·(id∗⊗X+⊗id∗)·(id∗⊗X+⊗id∗)·(min−→⊗min←−)


and−→max ·(id⊗−→max id∗)·(X+⊗id∗⊗id∗)·(X+⊗id∗⊗id∗)·(X+⊗id∗⊗id∗)·(id⊗min←−⊗id∗)·min←−(a) Show that these two knots are equivalent.(b) Indicate a sequence of the Turaev moves that transforms one prod-

uct into another.(c) Forget about the orientations and consider the corresponding un-

oriented tangles. Find a sequence of unoriented Turaev moves thattransforms one product into another.

(19) (Whitney trick) Show that the move FΩ1 in theframed Reidemeister Theorem 1.8.2 can be re-placed by the move depicted on the right.

(20) The group Zk+12 acts on oriented k-component links, changing the ori-

entation of each component and that of the ambient space. How manydifferent links are there in the orbit of an oriented Whitehead link underthis action?

Chapter 2

Knot invariants

A knot is a smooth embedding of the circle into 3-space. When such anembedding is continuously deformed, without running into self-intersections,the knot remains equivalent to itself.

To distinguish between (equivalent classes of) knots, it is natural to useknot invariants.

2.1. Definition and first examples

Let K be the set of all equivalence classes of knots.

Definition. A knot invariant is a function of a knot which takes equalvalues on equivalent knots. In other words, it is a mapping v : K → S fromK to a certain set S.

2.1.1. Crossing number. As we know, any knot can be represented by aplane diagram in an infinite number of ways.

Definition. The crossing number c(K) of a knot K is the minimal numberof crossing points in a plane diagram of K.

Exercise. Prove that if c(K) ≤ 2, then the knot K is trivial.

It follows that the minimal number of crossings required to draw a di-agram of a nontrivial knot is at least 3. A little later we will see that thetrefoil knot is indeed nontrivial.

Obviously, c(K) is a knot invariant taking values in the set of non-negative integers.

33

34 2. Knot invariants

2.1.2. Unknotting number. Another integer-valued invariant of a knotwhich admits a simple definition is the unknotting number.

Represent a knot by a plane diagram. The diagram can be transformedby plane isotopies, Reidemeister moves and crossing switches:

As we know, deformations of the first two kinds preserve the topologicaltype of the knot, and only crossing switches can change it.

Definition. The unknotting number u(K) of a knot K is the minimal num-ber of crossing changes in a plane diagram of K that convert it to a trivialknot, provided that plane isotopies and Reidemeister moves are allowed inany quantity.

Exercise. What is the unknotting number of the knots 31 and 83?

Finding the unknotting number, if it is greater than 1, is a difficult task;for example, the second question of the previous exercise was answered onlyin 1986 (by T. Kanenobu and H. Murakami).

2.1.3. Group of the knot. In general, the range of a knot invariant canbe an arbitrary set.

Definition. The group of the knot is the fundamental group of the com-plement to the knot in the ambient space: π(K) = π1(R

3 \K),

In this example, S is the set of isomorphism classes of groups. See, e.g.,[Rol, CrF] for a detailed discussion of the groups of knots.

Exercise. Prove that

(1) a knot is trivial if and only if its group is infinite cyclic,

(2) the group of the trefoil is generated by two elements x, y with onerelation x2 = y3,

(3) this group is isomorphic to the braid group B3 (in terms of x, y findanother pair of generators a, b that satisfy aba = bab).

2.2. Linking number

After the first general-purpose examples of knot invariants, discussed in theprevious section, here we want to introduce the linking number — we willsee later that this is the only Vassiliev invariant of degree 1.

The linking number of two spatial curves lk(A,B) provides an invariantof two-component links; its analog for knots (self-linking number) exists onlyin the case of framed knots.

2.2. Linking number 35

Intuitively, lk(A,B) is the number of times one curve winds aroundanother. To give a precise definition, choose an oriented diskDA immersed inspace so that its boundary is the curve A in agreement with the orientations.The linking number lk(A,B) is by definition equal to the intersection numberof DA and B. To find the intersection number, if necessary, make a smallperturbation of DA so it meets the curve B only at a finite number of pointsof transversal intersection. At each intersection point define the sign equalto ±1 depending on the orientations of DA, B and R3 at this point. Morespecifically, let (e1, e2) be a positive pair of tangent vectors to DA, while e3 apositively directed tangent vector to B at the intersection point; the sign isset to +1 iff the frame (e1, e2, e3) defines a positive orientation of R3. Thenthe linking number lk(A,B) is the sum of these signs over all intersectionpoints p ∈ DA ∩ B. One can prove that the result does not depend on thechoice of the surface DA and that lk(A,B) = lk(B,A).

Example: the two curves shown in the picture

have linking number equal to −1.

Given a plane diagram of the two-component link, there is a simplecombinatorial formula for the linking number. Let I the set of crossingpoints between A and B (crossing points of the separate components Aand B are here irrelevant). Then I is the disjoint union of two subsets IAB(overpasses of the component A) and IBA (overpasses of the component B).

2.2.1. Proposition.

lk(A,B) =∑

p∈IAB

w(p) =∑

p∈IBA

w(p) =1

2

∑

p∈Iw(p)

where w(p) is the local writhe of the crossing point.

Proof. Crossing switches at all points p ∈ IBA make the two componentsunlinked. Call the new curves A′ and B′, then lk(A′, B′) = 0. As can beseen from the pictures:


each crossing switch changes the linking number by −w where w is the localwrithe. Therefore, lk(A,B)−∑p∈IB

Aw(p) = 0, and the assertion follows.

Example: for these two curves both ways to compute the linking numbergive +1:

A

B

+

-

+

+

2.2.2. Analytial formulas. The linking number can be computed by an-alytical formulas in different ways. The most renowned one is the followingformula of Gauss (see [Spi] for a proof).

Theorem. Let A and B be two non-intersecting curves in R3, given by theirparameterizations α(u) and β(v) where α and β are smooth maps from thecircle S1 into 3-space. Then

lk(A,B) =1

4π

∫

S1×S1

(β(v)− α(u), du, dv)

|β(v)− α(u)|3 ,

where the parentheses in the numerator stand for the mixed product of 3vectors.

Geometrically, this formula computes the degree of the Gauss map fromA×B = S1×S1 to the 2-sphere S2, i.e. the number of times the normalizedvector connecting a point on A to a point on B goes around the sphere.

In relation to the material of this book another integral formula is muchmore relevant. It represents a simple version of the Kontsevich integral andwill be stated and proved in Chapter 8, see page 189.

2.3. Alexander–Conway polynomial 37

2.2.3. Self-linking. Let K be a framed knot and let K ′ be the knot ob-tained from K by a small shift in the direction of the framing:

Definition. The self-linking number of K is the linking number of K andK ′.

Note in passing that the linking number is the same if K is shifted inthe direction, opposite to the framing.

Proposition. The self-linking number of a framed knot given by a diagramD with blackboard framing is equal to the total writhe of the diagram D.

Proof. Indeed, in the case of blackboard framing, the only crossings be-tween K and K ′ occur at the crossing points of K. The vicinity of eachcrossing point looks like

K

K’K

K’

Here we see one overpass of K over K ′ with the local writhe equal to thelocal writhe of the initial crossing point for the knot K itself. Therefore,the claim follows from the combinatorial formula for the linking number(Proposition 2.2.1).

2.3. Alexander–Conway polynomial

Below, we will be interested in algebraic operations (addition and, less fre-quently, multiplication) on the sets of invariants, therefore we will usuallyrestrict ourselves to the case when the invariants take values in a commu-tative ring. Of particular importance in knot theory are polynomial knotinvariants taking values in the rings of polynomials (or Laurent polynomi-als1) in one or two variables, usually with integer coefficients.

Historically, the first polynomial invariant for knots was the Alexanderpolynomial A(K) introduced in 1928 [Al]. See [CrF, Lik, Rol] for a discus-sion of the beautiful topological theory related to Alexander’s polynomial.In 1970 J. Conway [Con] found a simple recursive construction of an invari-ant polynomial C(K) which differs from Alexander’s polynomial only by achange of variables, namely, A(K) = C(K) |t7→t1/2−t−1/2 . In this book, wewill only use Conway’s normalization, but sometimes refer to the polynomialas Alexander–Conway polynomial. Conway’s definition, given in terms ofplane diagrams, relies on crossing point resolutions that may take a knotdiagram into a link diagram, therefore it must be given for links.

1A Laurent polynomial in x is a polynomial in x and x−1.


2.3.1. Definition. The Conway polynomial C is an invariant of orientedlinks (therefore, in particular, an invariant of oriented knots) defined by thetwo properties:

C( )

= 1,

tC( )

= C( )

− C( )

.

Here stands for the unknot (trivial knot) while the three pictures

in the second line mean three knots that are identical everywhere except forthe fragments shown. The second relation is referred to as Conway’s skeinrelation. Skein relations are equations between the values of some quantityon knots (links, etc.) represented by diagrams that differ from each otherby local changes near a crossing point, provide a convenient way to workwith knot invariants.

It is not quite trivial to prove the existence of an invariant satisfyingthis definition, but as soon as this fact is established, the computation ofthe Conway polynomial is fairly easy.

2.3.2. Example.

C( )

=1

tC( )

− 1

tC( )

= 0,

because the two knots on the right are equivalent (both are trivial).

2.3.3. Example.

C( )

= C( )

− tC( )

= C( )

− tC( )

= −t

2.3.4. Example.

C( )

= C( )

− tC( )

= C( )

− tC( )

= 1 + t2

2.4. Jones polynomial 39

2.3.5. The values of the Conway polynomial on table knots with up to 8crossings are given in Table 1. Note that the Conway polynomial of theinverse knot K∗ and the mirror knot K is always the same as for the initialknot K.

K C(K) K C(K) K C(K)

31 1 + t2 76 1 + t2 − t4 811 1− t2 − 2t4

41 1− t2 77 1− t2 + t4 812 1− 3t2 + t4

51 1 + 3t2 + t4 81 1− 3t2 813 1 + t2 + 2t4

52 1 + 2t2 82 1− 3t4 − t6 814 1− 2t4

61 1− 2t2 83 1− 4t2 815 1 + 4t2 + 3t4

62 1− t2 − t4 84 1− 3t2 − 2t4 816 1 + t2 + 2t4 + t6

63 1 + t2 + t4 85 1− t2 − 3t4 − t6 817 1− t2 − 2t4 − t671 1 + 6t2 + 5t4 + t6 86 1− 2t2 − 2t4 818 1 + t2 − t4 − t672 1 + 3t2 87 1 + 2t2 + 3t4 + t6 819 1 + 5t2 + 5t4 + t6

73 1 + 5t2 + 2t4 88 1 + 2t2 + 2t4 820 1 + 2t2 + t4

74 1 + 4t2 89 1− 2t2 − 3t4 − t6 821 1− t475 1 + 4t2 + 2t4 810 1 + 3t2 + 3t4 + t6

Table 1. Conway polynomials of table knots

For every n, the coefficient cn of tn in C is a numeric invariant of theknot.

2.4. Jones polynomial

The invention of the Jones polynomial [Jo1] in 1985 produced a genuinerevolution in knot theory. The original construction of V. Jones was givenin terms of state sums and von Neumann algebras. It was soon noted,however, that the Jones polynomial can be defined by skein relations, in thespirit of Conway’s definition 2.3.1.

Instead of bluntly giving the corresponding formal equations, we willexplain, following L. Kauffman [Ka6], how this definition could be invented.Like with Conway’s polynomial, the construction given below requires thatwe consider invariants on the totality of all links, not only knots, becausethe transformations used may turn a knot diagram into a many-componentlink diagram.

Suppose that we are looking for an invariant of non-oriented links, de-noted by angular brackets, that has a prescribed behavior with respect tothe resolution of diagram crossings and addition of a disjoint copy of the


unknot:

〈〉 = a 〈〉+ b 〈〉,

〈L ⊔ 〉 = c 〈L 〉,

where a, b and c are certain fixed coefficients.

In order that the bracket 〈 , 〉 be a link invariant, it must be stable underthe three Reidemeister moves Ω1, Ω2, Ω3 (see Sec. 1.3).

2.4.1. Exercise. Show that the bracket 〈, 〉 is Ω2-invariant if and only ifb = a−1 and c = −a2 − a−2.

2.4.2. Exercise. Prove that Ω2-invariance implies Ω3-invariance.

2.4.3. Exercise. Suppose that b = a−1 and c = −a2 − a−2. Check thatthe behavior of the bracket with respect to the first Reidemeister move isdescribed by the equations

〈〉 = −a−3 〈〉,

〈〉 = −a3 〈〉.

In the assumptions b = a−1 and c = −a2 − a−2 the bracket polynomial〈L〉 normalized by the initial condition

〈〉 = 1

is referred to as the Kauffman bracket of L. We see that the Kauffmanbracket changes only under the addition (or deletion) of a crossing, andthis change depends on the local writhe of the crossing. It is therefore easyto write a formula for a quantity that would be invariant under all threeReidemeister moves:

J(L) = (−a)−3w〈L〉,where w is the total writhe of the diagram (the difference between the num-ber of positive and negative crossings).

This quantity J(L) is a Laurent polynomial called the Jones polynomialin a-normalization. The more standard t-normalization is obtained by thesubstitution a = t−1/4. Note that the Jones polynomial is an invariant of anoriented link, although in its definition we use Kauffman’s bracket which isdetermined by the diagram without orientation.

2.4.4. Exercise. Check that the Jones polynomial is uniquely determinedby the skein relation

t−1J( )− tJ( ) = (t1/2 − t−1/2)J( ) (1)

2.5. Algebra of knot invariants 41

and the initial condition

J( ) = 1. (2)

2.4.5. Example. Let us compute the value of the Jones polynomial on thepositive (left) trefoil. The calculation requires several steps, each consistingof one application of the rule (1) and some applications of rule (2) and/orusing the results of the previous steps. We leave the details to the reader.

(i) J( )

= −t1/2 − t−1/2.

(ii) J( )

= −t1/2 − t5/2.

(iii) J( )

= −t−5/2 − t−1/2.

(iv) J

( )= −t−4 + t−3 + t−1.

2.4.6. Exercise. Repeat the previous calculation for the right trefoil andprove that J(31) = t+ t3 − t4.

We see that Jones’ polynomial J can tell apart two knots which Conway’spolynomial C cannot. This does not mean, however, that J is stronger thanC. There are pairs of knots, for example, K1 = 1071, K2 = 10104 such thatJ(K1) = J(K2), but C(K1) 6= C(K2) (see, e.g. [Sto2]).

2.4.7. The values of Jones polynomial on table knots with up to 8 crossingsare given in Table 2. Note that the Jones polynomial does not changed whenthe the knot is inversed; its behavior under mirror reflection is described inExercise 8.

2.5. Algebra of knot invariants

Knot invariants with values in a given commutative ring F form an algebraI over that ring with respect to usual pointwise operations on functions

(f + g)(K) = f(K) + g(K),(1)

(fg)(K) = f(K)g(K).(2)

As a vector space2, I is dual to the space of knots X introduced inSection 1.7. Indeed, there is a natural bijection between all functions on agiven set (like K) and linear functions on the linear space freely generatedby this set (in our example, X ):

I = HomF(X ,F).

2More exactly, I is a module over F, but we will call free modules vector spaces, by abuse oflanguage.


31 −t−4 + t−3 + t−1

41 t−2 − t−1 + 1− t+ t2

51 −t−7 + t−6 − t−5 + t−4 + t−2

52 −t−6 + t−5 − t−4 + 2t−3 − t−2 + t−1

61 t−4 − t−3 + t−2 − 2t−1 + 2− t+ t2

62 t−5 − 2t−4 + 2t−3 − 2t−2 + 2t−1 − 1 + t63 −t−3 + 2t−2 − 2t−1 + 3− 2t+ 2t2 − t371 −t−10 + t−9 − t−8 + t−7 − t−6 + t−5 + t−3

72 −t−8 + t−7 − t−6 + 2t−5 − 2t−4 + 2t−3 − t−2 + t−1

73 t2 − t3 + 2t4 − 2t5 + 3t6 − 2t7 + t8 − t974 t− 2t2 + 3t3 − 2t4 + 3t5 − 2t6 + t7 − t875 −t−9 + 2t−8 − 3t−7 + 3t−6 − 3t−5 + 3t−4 − t−3 + t−2

76 −t−6 + 2t−5 − 3t−4 + 4t−3 − 3t−2 + 3t−1 − 2 + t77 −t−3 + 3t−2 − 3t−1 + 4− 4t+ 3t2 − 2t3 + t4

81 t−6 − t−5 + t−4 − 2t−3 + 2t−2 − 2t−1 + 2− t+ t2

82 t−8 − 2t−7 + 2t−6 − 3t−5 + 3t−4 − 2t−3 + 2t−2 − t−1 + 183 t−4 − t−3 + 2t−2 − 3t−1 + 3− 3t+ 2t2 − t3 + t4

84 t−5 − 2t−4 + 3t−3 − 3t−2 + 3t−1 − 3 + 2t− t2 + t3

85 1− t+ 3t2 − 3t3 + 3t4 − 4t5 + 3t6 − 2t7 + t8

86 t−7 − 2t−6 + 3t−5 − 4t−4 + 4t−3 − 4t−2 + 3t−1 − 1 + t87 −t−2 + 2t−1 − 2 + 4t− 4t2 + 4t3 − 3t4 + 2t5 − t688 −t−3 + 2t−2 − 3t−1 + 5− 4t+ 4t2 − 3t3 + 2t4 − t589 t−4 − 2t−3 + 3t−2 − 4t−1 + 5− 4t+ 3t2 − 2t3 + t4

810 −t−2 + 2t−1 − 3 + 5t− 4t−2 + 5t3 − 4t4 + 2t5 − t6811 t−7 − 2t−6 + 3t−5 − 5t−4 + 5t−3 − 4t−2 + 4t−1 − 2 + t812 t−4 − 2t−3 + 4t−2 − 5t−1 + 5− 5t+ 4t2 − 2t3 + t4

813 −t−3 + 3t−2 − 4t−1 + 5− 5t+ 5t2 − 3t3 + 2t4 − t5814 t−7 − 3t−6 + 4t−5 − 5t−4 + 6t−3 − 5t−2 + 4t−1 − 2 + t815 t−10 − 3t−9 + 4t−8 − 6t−7 + 6t−6 − 5t−5 + 5t−4 − 2t−3 + t−2

816 −t−6 + 3t−5 − 5t−4 + 6t−3 − 6t−2 + 6t−1 − 4 + 3t− t2817 t−4 − 3t−3 + 5t−2 − 6t−1 + 7− 6t+ 5t2 − 3t3 + t4

818 t−4 − 4t−3 + 6t−2 − 7t−1 + 9− 7t+ 6t2 − 4t3 + t4

819 t3 + t5 − t8820 −t−5 + t−4 − t−3 + 2t−2 − t−1 + 2− t821 t−7 − 2t−6 + 2t−5 − 3t−4 + 3t−3 − 2t−2 + t−1

Table 2. Jones polynomials of table knots

This bijection is, however, only a linear isomorphism; the algebraic struc-tures of X and I are different. Later (in Chapter 4) we will enrich the al-gebraic structure of both X and I thus obtaining a pair of mutually dualbialgebras.

2.6. Quantum invariants 43

2.6. Quantum invariants

This section concerns material which is less elementary than the rest of thebook. We are not going to develop a full theory of quantum groups andcorresponding invariants here, confining ourselves to some basic ideas whichcan be understood without going deep into complicated formalities. Thereader will see that it is possible to use quantum invariants without evenknowing what a quantum group is!

2.6.1. The discovery of the Jones polynomial inspired many people to searchfor other skein relations compatible with Reidemeister moves and thus defin-ing knot polynomials. This lead to the introduction of the HOMFLY ([HOM,PT]) and Kauffman’s ([Ka3, Ka4]) polynomials. It soon became clear thatall these polynomials are the first members of a vast family of knot invariantscalled quantum invariants.

The original idea of quantum invariants (in the case of 3-manifolds) wasproposed by E. Witten in the famous paper [Wit1]. Witten’s approach com-ing from physics was not completely justified from the mathematical view-point. The first mathematically impeccable definition of quantum invariantsof links and 3-manifolds was given by Reshetikhin and Turaev [RT1, Tur2],who used in their construction the notion of quantum groups freshly intro-duced by V. Drinfeld in [Dr4] (see also [Dr3]) and M. Jimbo in [Jimb].In fact, a quantum group is not a group at all. Instead, it is a family ofalgebras, more precisely, of Hopf algebras (see Appendix A.2.20), depend-ing on a complex parameter q and satisfying certain axioms. The quantumgroup Uqg of a semisimple Lie algebra g is a remarkable deformation of theuniversal enveloping algebra (see Appendix A.1.7) of g (corresponding tothe value q = 1) in the class of Hopf algebras.

In this section, we show how the Jones polynomial J can be obtainedby the techniques of quantum groups, following the approach of Reshetikhinand Turaev. It turns out that J coincides, up to normalization, with thequantum invariant corresponding to the Lie algebra g = sl2 in its standardtwo-dimensional representation (see Appendix A.1.1). Later in the book,we will sometimes refer to the ideology exemplified in this section. Fordetailed expositions of quantum groups, we refer the interested reader to[Jan, Kas, KRT].

2.6.2. Let g be a semisimple Lie algebra and let V be its finite dimensionalrepresentation. One can think about V as a representation of the universalenveloping algebra of g (see Appendix, p. 336). It is remarkable that thisrepresentation can also be deformed with parameter q to a representationof the quantum group Uqg. The vector space V remains the same, but the


action now depends on q. For a generic value of q all irreducible representa-tions of Uqg can be obtained in this way. However, when q is a root of unitythe representation theory is different and resembles the representation the-ory of g in finite characteristic. It can be used to derive quantum invariantsof 3-manifolds. For the purposes of knot theory it is enough to use genericvalues of q.

2.6.3. One of the most important features of quantum groups is that everyrepresentation gives rise to a solution R the quantum Yang–Baxter equation

(R⊗ idV )(idV ⊗R)(R⊗ idV ) = (idV ⊗R)(R⊗ idV )(idV ⊗R)

where R (R-matrix ) is an invertible linear operator R : V ⊗ V → V ⊗ V ,and both sides of the equation are understood as linear transformationsV ⊗ V ⊗ V → V ⊗ V ⊗ V .

Exercise. Given an R-matrix, construct a representation of the braidgroup Bn in the space V ⊗n.

There is a procedure to construct an R-matrix associated with a re-presentation of a Lie algebra. We are not going to describe it in general,confining ourselves just to one example: the Lie algebra g = sl2 and its stan-dard two dimensional representation V . In this case the associated R-matrixhas the form

R :

e1 ⊗ e1 7→ q1/4e1 ⊗ e1e1 ⊗ e2 7→ q−1/4e2 ⊗ e1e2 ⊗ e1 7→ q−1/4e1 ⊗ e2 + (q1/4 − q−3/4)e2 ⊗ e1e2 ⊗ e2 7→ q1/4e2 ⊗ e2

for an appropriate basis e1, e2 of the space V . The inverse of R (we willneed it later) is given by the formulas

R−1 :

e1 ⊗ e1 7→ q−1/4e1 ⊗ e1e1 ⊗ e2 7→ q1/4e2 ⊗ e1 + (−q3/4 + q−1/4)e1 ⊗ e2e2 ⊗ e1 7→ q1/4e1 ⊗ e2e2 ⊗ e2 7→ q−1/4e2 ⊗ e2

2.6.4. Exercise. Check that this operator R satisfies the quantum Yang-Baxter equation.

2.6.5. The general procedure of constructing quantum invariants is orga-nized as follows (see details in [Oht1]). Consider a knot diagram in theplane and take a generic horizontal line. To each intersection point of theline with the diagram we assign either the representation space V or its dualV ∗ depending on whether the orientation of the knot at this intersection isdirected upwards or downwards. Then take the tensor product of all such


spaces over the whole horizontal line. If the knot diagram does not intersectthe line then the corresponding vector space is the ground field C.

A portion of a knot diagram between two such horizontal lines representsa tangle T (see the general definition in Sec. 1.5). With T we associatea linear transformation θfr(T ) from the vector space corresponding to thebottom of T to the vector space corresponding to the top of T . The followingtwo properties hold for the linear transformation θfr(T ).

• θfr(T ) is an invariant of the isotopy class of the tangle T• θfr(T1 · T2) = θfr(T1) θfr(T2)

V

V

T1

T2

VV

V V

V

VVV *

*V ∗ ⊗ V ⊗ V ⊗ V

V ⊗ V ∗ ⊗ V ⊗ Vθfr(T1)

OO

V ⊗ Vθfr(T2)

OOθfr(T1·T2)

hh

Now we can define a knot invariant θfr(K) regarding the knot K as a tanglebetween the two lines below and above K. In this case θfr(K) would be alinear transformation from C to C, i.e. multiplication by a number. Since ourlinear transformations depend the parameter on q, this number is actuallya function of q.

2.6.6. Because of the multiplicativity property θfr(T1 · T2) = θfr(T1) θfr(T2) it is enough to define θfr(T ) only for elementary tangles T suchas a crossing, a minimum or a maximum point. This is precisely wherequantum groups come in. Given a quantum group Uqg and its finite dimen-sional representation V , one can associate certain linear transformationswith elementary tangles in a consistent way. The R-matrix appears here asthe linear transformation corresponding to a positive crossing, while R−1

corresponds to a negative crossing. Of course, for a trivial tangle consistingof a single string connecting the top and bottom, the corresponding linearoperator should be the identity transformation. So we have the followingcorrespondence valid for all quantum groups.

V

V

VxidV

VV *

V

*V ∗xidV ∗

V ∗


V

V

V

V V ⊗ VxR

V ⊗ VV

V

V

V V ⊗ VxR−1

V ⊗ V

Using this we can easily check that the invariance of a quantum invariantunder the third Reidemeister move is nothing else than the quantum Yang–Baxter equation:

V ⊗ V ⊗ VR⊗idV

x

V ⊗ V ⊗ VidV ⊗R

x

V ⊗ V ⊗ VR⊗idV

x

V ⊗ V ⊗ V

VV

V VV

V

V VV

V V V

=

VV

V VV

V

V VV

V V V

V ⊗ V ⊗ VxidV ⊗R

V ⊗ V ⊗ VxR⊗idV

V ⊗ V ⊗ VxidV ⊗R

V ⊗ V ⊗ V

(R⊗ idV )(idV ⊗R)(R⊗ idV ) = (idV ⊗R)(R⊗ idV )(idV ⊗R)

Similarly the assignment of mutually inverse operators (R and R−1)to positive and negative crossings implies the invariance under the secondReidemeister move. (The first Reidemeister move is treated in Exercise 2abelow.)

To complete the construction of a quantum group invariant we shouldassign appropriate operators to the minimum and maximum points. Thisdepends on all involved data: the quantum group, the representation andthe R-matrix. For the quantum group Uqsl2, its standard two dimensionalrepresentation V and the R-matrix chosen in 2.6.3 these operators are:

V

V* V ∗ ⊗ V

C

OO q−1/2e1 ⊗ e1 + q1/2e2 ⊗ e2

1_

OO

V

V * V ⊗ V ∗

C

OO e1 ⊗ e1 + e2 ⊗ e2

1_

OO


V

V *

C

V ⊗ V ∗

OO q1/2

e1 ⊗ e1_

OO 0

e1 ⊗ e2_

OO 0

e2 ⊗ e1_

OOq−1/2

e2 ⊗ e2_

OO

V

V*

C

V ∗ ⊗ V

OO 1

e1 ⊗ e1_

OO 0

e1 ⊗ e2_

OO 0

e2 ⊗ e1_

OO 1

e2 ⊗ e2_

OO

where e1, e2 is a basis of V ∗ dual to the basis e1, e2 of the space V .

2.6.7. Example. Let us compute the sl2-quantum invariant of the unknot.Represent the unknot as a product of two tangles and compute the compo-sition of the corresponding transformations

V

V*

C

V ∗ ⊗ V

OO

C

OO

q−1/2

︸︷︷︸ + q1/2︸︷︷︸

_

OO

_

OO

︷︸︸︷q−1/2e1 ⊗ e1 +

︷︸︸︷q1/2e2 ⊗ e2︸︷︷︸

1_

OO

So θfr(unknot) = q1/2 + q−1/2. Therefore, to normalize our invariant sothat its value on the unknot would be equal to 1, we must divide θfr(·) by

q1/2 + q−1/2.


2.6.8. Example. Let us compute the quantum invariant for the left trefoil.Represent the diagram of the trefoil as follows.

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

V

*

*

*

*

*

*

*

*

*

*

Cx

V ∗ ⊗ Vx

V ∗ ⊗ V ⊗ V ⊗ V ∗xidV ∗⊗R−1⊗idV ∗

V ∗ ⊗ V ⊗ V ⊗ V ∗xidV ∗⊗R−1⊗idV ∗

V ∗ ⊗ V ⊗ V ⊗ V ∗xidV ∗⊗R−1⊗idV ∗

V ∗ ⊗ V ⊗ V ⊗ V ∗x

V ⊗ V ∗x

C

Two maps at the bottom send 1 ∈ C into the tensor

1 7→ q−1/2e1 ⊗ e1 ⊗ e1 ⊗ e1 + q−1/2e1 ⊗ e1 ⊗ e2 ⊗ e2

+ q1/2e2 ⊗ e2 ⊗ e1 ⊗ e1 + q1/2e2 ⊗ e2 ⊗ e2 ⊗ e2 .

Then appying R−3 to two tensor factors in the middle we get

q−1/2e1 ⊗(q−3/4e1 ⊗ e1

)⊗ e1

+q−1/2e1 ⊗((−q9/4 + q5/4 − q1/4 + q−3/4

)e1 ⊗ e2

+(−q7/4 − q3/4 − q−1/4

)e2 ⊗ e1

)⊗ e2

+q1/2e2 ⊗((q7/4 − q3/4 + q−1/4

)e1 ⊗ e2 +

(−q5/4 + q1/4

)e2 ⊗ e1

)⊗ e1

+q1/2e2 ⊗(q−3/4e2 ⊗ e2

)⊗ e2 .

2.7. Two-variable link polynomials 49

Finally, the two maps at the top contract the whole tensor into a number

θfr(31) = q−1/2q−3/4q1/2 + q−1/2(−q9/4 + q5/4 − q1/4 + q−3/4

)q−1/2

+q1/2(−q5/4 + q1/4

)q1/2 + q1/2q−3/4q−1/2

= 2q−3/4 − q5/4 + q1/4 − q−3/4 + q−7/4 − q9/4 + q5/4

= q−7/4 + q−3/4 + q1/4 − q9/4

Dividing by the normalizing factor q1/2 + q−1/2 we get

θfr(31)

q1/2 + q−1/2= q−5/4 + q3/4 − q7/4 .

The invariant θ(K) we just exemplified does not change under the secondand third Reidemeister moves. However it changes under the first Reide-meister move and thus is a framing-dependent invariant. One can deframeit according to the formula

θ(K) = q−c·w(K)

2 θfr(K) ,

where w(K) is the writhe of the knot diagram and c is the quadratic Casimirnumber (see Appendix A.1.1) defined by the original Lie algebra g andits representation. For sl2 and the standard 2-dimensional representationc = 3/2. The writhe of the left trefoil in our example equals −3. So for theunframed quantum invariant we have

θ(31)

q1/2 + q−1/2= q9/4

(q−5/4 + q3/4 − q7/4

)= q + q3 − q4 .

The substitution q = t−1 gives the Jones polynomial t−1 + t−3 − t−4.

2.7. Two-variable link polynomials

2.7.1. HOMFLY polynomial. There is an unframed link invariant, HOM-FLY polynomial, P (L) which is a Laurent polynomial in two variables l andm with integer coefficients satisfying the skein relation and the initial con-dition

lP ( ) − l−1P ( ) = mP ( ) ; P ( ) = 1 .

The existence of such an invariant is a difficult theorem. It was discoveredsimultaneously and independently by five groups of authors [HOM, PT](see also [Lik]). The HOMFLY polynomial is equivalent to the collectionof quantum invariants associated with the Lie algebra slN and its standardN -dimensional representation for all values of N .


Important properties of the HOMFLY polynomial are contained in thefollowing exercises.

(1) (a) Prove the uniqueness of such an invariant. Or, sim-ply speaking, prove that the relation above are suf-ficient to compute the HOMFLY polynomial.

(b) Compute HOMFLY polynomial for the knots31, 41, 51, and 52.

(c) (A. Sossinsky [Sos]) Compare the HOMFLYpolynomials of two knots on the right.

(2) Prove that the HOMFLY polynomial of a knot is preserved whenthe knot orientation is reversed.

(3) (W. B. R. Lickorish [Lik]) Prove that

(a) P (L) = P (L), where L is the mirror reflection

of L, and P (L) is the polynomial obtainedfrom P (L) by substitution l−1 instead of l;

(b) P (K1#K2) = P (K1) · P (K2);

(c) P (L1 ⊔ L2) =l − l−1

m· P (L1) · P (L2), where

L1 ⊔ L2 means the separated union of links;

88 =

10129 =

(d) P (88) = P (10129).These knots may be distinguished by the Kauffman two-variablepolynomial defined below.

2.7.2. Two-variable Kauffman polynomial. In [Ka4] L. Kauffman foundanother invariant Laurent polynomial of two variables a and z. Its framedversion is defined for unoriented links diagrams with the blackboard framingvia skein relations

Λfr( ) − Λfr( ) = z(Λfr( ) − Λfr( )

),

Λfr( ) = aΛfr( ) ,

and the initial condition that Λfr(unknot) = 1.

Its oriented version can be defined by

Λ(L) := a−writhe(L)Λfr(L)

for any oriented (but now unframed) link diagram L. It turns out that thispolynomial is equivalent to the collection of the quantum invariants associ-ated with the Lie algebra soN and its standardN -dimensional representationfor all values of N (see [Tur3]).

2.7. Two-variable link polynomials 51

Like in the previous section, we conclude with a series of exercises withadditional information on the Kauffman polynomial.

(1) Prove that the defining relations are sufficient to compute the Kauff-man polynomial.

(2) Compute Kauffman polynomial for the knots 31, 41, 51, and 52.

(3) Prove that the Kauffman polynomial of a knot is preserved whenthe knot orientation is reversed.

(4) (W. B. R. Lickorish [Lik]) Prove that

(a) Λ(L) = Λ(L), where L is the mirror reflec-

tion of L, and Λ(L) is the polynomial obtainedfrom Λ(L) by substitution a−1 instead of a;

(b) Λ(K1#K2) = Λ(K1) · Λ(K2);

(c) Λ(L1⊔L2) =((a+a−1)z−1−1

)·Λ(L1)·Λ(L2),

where L1 ⊔ L2 means the separated union oflinks;

11255 =

11257 =

(d) Λ(11255) = Λ(11257);(these knots may be distinguished either by the HOMFLY orby the Conway polynomials)

(e) Λ(L∗) = a4lk(K,L−K)Λ(L), where the link L∗ is obtained froman oriented link L by reversing the orientation of one compo-nent K.

2.7.3. Comparison between polynomials. Let us say that an invariantI1 dominates an invariant I2, if the equality I1(K1) = I1(K2) for two knotsK1 and K2 always implies the equality I2(K1) = I2(K2). Denoting thisrelation by arrows, we have the following comparison chart:

HOMFLY

Alexander(=Conway)

Kauffman

Jones

(the absence of an arrow between the two invariants means that neither ofthem dominates the other).

Exercise. Collect, in the previous text of the book, all the facts suffi-cient to justify this chart.


Exercises

(1) Prove that Conway’s and Jones’ polynomials of a knot are preservedwhen the knot orientation is reversed.

(2) Let θ(·) and θfr(·) be the quantum invariants constructed in 2.6.3 and2.6.6 for the Lie algebra sl2 and its standard 2-dimensional representa-tion.(a) Prove the following dependence of θfr(·) on the first Reidemeister

move

θfr( ) = q3/4θfr( ) .

(b) Prove that this θ(·) does not change under the first Reidemeistermove.

(c) Compute the value θ(41).(d) Show that R-matrix from page 44 satisfies the equation

q1/4R− q−1/4R−1 = (q1/2 − q−1/2)idV ⊗V .

(e) Prove that θfr(·) satisfies the skein relation

q1/4θfr( ) − q−1/4θfr( ) = (q1/2 − q−1/2)θfr( ) .

(f) Prove that θ(·) satisfies the skein relation

qθ( ) − q−1θ( ) = (q1/2 − q−1/2)θ( ) .

(g) For any link L with k components prove that

θfr(L) = (−1)k(q1/2 + q−1/2) · 〈L〉∣∣∣a=−q1/4

,

where 〈·〉 is the Kauffman bracket defined on page 40.

(3) Prove that C(K1#K2) = C(K1) · C(K2), where C denotes the Conwaypolynomial.

(4) Prove that J(K1#K2) = J(K1) · J(K2), where J denotes the Jonespolynomial.

(5) Prove that for any knot K the Conway polynomial C(K) is an evenpolynomial in t and its free term equals 1:

C(K) = 1 + c2(K)t2 + c4(K)t4 + c6(K)t6 + . . .

(6) Prove that for a link L with an odd number of components, J(L) isa polynomial in t and t−1, and for a link L with an even number ofcomponents J(L) = t1/2 · (a polynomial in t and t−1).

Exercises 53

(7) Prove that for a link L with k components J(L)∣∣t=1

= (−2)k−1. In

particular, J(K)∣∣t=1

= 1 for a knot K.

(8) For the mirror reflection L of a link L prove that J(L) is obtained fromJ(L) by substituting t−1 instead of t.

(9) For the link L∗ obtained from an oriented link L by reversing the orien-

tation of one of its components K prove that J(L∗) = t−3lk(K,L−K)J(L).

(10) Compute the Conway polynomials of the Conway and the Kinoshita–Teresaka knots from page 226.

(11) ∗ Find a non trivial knot K with J(K) = 1.

(12) (L. Kauffman [Ka6], K. Murasugi [Mur], M. Thistlethwaite [Th]).Prove the for a reduced alternating knot diagram K (Sec.1.3.3) thenumber of crossings is equal to span(J(K)), that is the difference be-ween the maximal and minimal degrees of t in the Jones polynomialJ(K). (Remark. This exercise is not difficult, although it solves a onehundred years old conjecture of Tait. Anyway, the reader can find asimple solution in [Tur1].)

Chapter 3

Finite type invariants

In this chapter we introduce the main protagonist of this book — finite type(Vassiliev) knot invariants. This is done by considering knots as points inthe (infinite-dimensional) space of smooth immersions of the circle into 3-space and studying the prolongation of knot invariants to the immersionswhich are not embeddings (singular knots).

After that, we introduce our main combinatorial object: chord diagrams,to which the book owes its title. Chord diagrams serve as a means to describethe symbols (highest parts) of the Vassiliev invariants.

Then we prove that classical invariant polynomials are all in a sensefinite type, and finally explain a simple straightforward way to calculate thevalues of Vassiliev invariants on any given knot and give a table of basicVassiliev invariants up to degree 5.

3.1. Discriminant approach

Recall that a knot is a smooth embedding of the circle S1 into 3-space R3. Asmooth isotopy in the class of embeddings does not change the equivalenceclass of a knot. To change the equivalence class, one must make one orseveral crossing changes.

Doing a crossing change continuously, one must pass through a mapfrom S1 to R3 which is not an embedding, but an immersion S1 → R3 witha double point:

55

56 3. Finite type invariants

At this moment, it is quite instructive to imagine simultaneously thetransformations of a closed curve in 3-space (the physical space in the ter-minology of Arnold [Ar3]) and corresponding paths in the space of mapsfrom S1 to R3 (the functional space). Here is a typical picture:

Σ

wall

1K

K 2

Path in the functional space

Isotopy of a knot

The disk on the left is the space Imm of immersions S1 → R3, the dashedlines are the discriminant Σ consisting of immersions which are not embed-dings, and the connected components of the complement Emb = Imm \ Σcorrespond to equivalence classes of knots.

Remark. In the original Vassiliev’s approach, he considered the spaceof all smooth maps S1 → R3 with both local and non-local singularitiesallowed:

Local singularities Non-local singularities

This is important to develop the general theory: the space of all maps islinear and hence contractible, therefore one can apply Alexander duality etc.(see details in [Va3]). However, it is clear that, to transform any knot intoany other, it is enough to pass only through immersions with one transversalself-intersection, so in this elementary introduction we can confine ourselveswith the class of immersions.

Embeddings and immersions from S1 to R3 constitute two infinite-dimen-sional manifolds, Emb ⊂ Imm. While the entire space Imm is connected,the subspace Emb is not, and its connected components are nothing but thetopological types of knots. The type can only change when the point in thespace Imm passes through a wall separating two connected components.

A knot invariant is the same thing as a function constant on the con-nected components of the space Emb. When a point in Imm representinga knot, goes through a wall, the invariant experiences a jump. Vassiliev’soriginal idea was to prolong knot invariants from Emb to Σ assigning to apoint p ∈ Σ the value of the corresponding jump when one passes from the

3.1. Discriminant approach 57

negative side to the negative side of Σ. This construction only makes sensefor generic points of the discriminant which correspond to knots with onesimple double point, and it relies on the fact that at these generic points thediscriminant is naturally cooriented.

Definition. A point p ∈ im(f) ⊂ R3 is a simple double point of f if itspreimage under f consists of two values t1 and t2 and the two tangentvectors f ′(t1) and f ′(t2) are non-collinear. Geometrically, this means that,in a neighborhood of the point p, the curve consists of two branches withdifferent tangents.

Simple double point

Smooth singular maps having one simple double point form a subman-ifold of codimension 1 in the space of all immersions, while for other typesof singularities the codimension is greater. They form the principal stratumof the discriminant Σ.

Definition. A hypersurface in a real manifold is said to be coorientable if ishas a non-zero section of its normal bundle, i.e. if there exists a continuousvector field which is not tangent to the hypersurface at any point and doesnot vanish anywhere. The only thing we want to know of this vector feld isthe side of the hypersurface it points to. The coorientation of a coorientablehypersurface is a choice of one of the two such possibilities.

The figure below on the left illustrates this notion. The figure on theright shows the simplest example of a hypersurface that has no coorientation— the Mobius band in R3.

Coorientation Mobius band

A crucial observation is that the (open) hypersurface in Imm consistingof all knots with exactly one simple double point is coorientable. A smallshift off the discriminant is said to be positive if the local writhe at thepoint in question is +1, and negative, if it is −1 (see page 19). Each of theseresolutions is well defined (does not depend on the plane projection used


to express this relation). To make sure of this fact, the reader is invitedto make a physical model of a crossing with the help of two sharpened (i.e.oriented) pencils and look at them from one side, then from the other.

3.1.1. While an arbitrary path from one knot to another can be effectuatedby a path that goes only through the generic points of the discriminant, itturns out to be very important to understand what happens near the pointswhere the discriminant has self-intersections. The simplest self-intersectionoccurs at the points where the corresponding singular knot has two simpledouble points:

d c

b a

Σ

Σ

a-b

b-d

c-d

a-c

NENW

SESW

The figure on the left shows the neighborhood of a self-intersection in thefunctional space; Σ is the dicriminant, arrows show its coorientation, the fatgrey points are typical points of the corresponding regions, and the lettersa, b, c, d, as well as their differences are the values of some knot invariant inthese regions. The figure on the right displays sample knots close to a knotwith two self-intersections in accordance with the left-hand picture. Notethat among the four non-singular knots which are present, one (NE) is atrefoil, while the other three (NW, SW, SE) are trivial knots and thereforebelong to the same global component of the space Imm \ Emb: the localconnected components shown in the figure are joined together somewherefar away.

We see that the difference of values of an invariant along one componentof the discriminant (a − b) − (c − d) is equal to the difference of its valueson another component (a − c) − (b − d). Therefore the invariant can beprolonged to the central point of the picture by the value a− b− c+ d.

This observation is valid for knots with any number of distinct simpledouble points. Let Kn denote the set of equivalence classes of singular knotswith n double points and no other singularities. By definition, K0 = K is

the set of all non-singular knots. Put K = ∪n≥0Kn.

Definition. (Vassiliev’s extension of knot invariants.) Let G be an Abelian

group. Given a knot invariant v : K → G, we define its extension K → G,

3.2. Definition of Vassiliev invariants 59

denoted by the same letter v, according to the rule

(1) v( ) = v( )− v( ),

known as Vassiliev’s skein relation.

The right hand side of Vassiliev’s skein relation refers to the two resolu-tions of the double point — positive and negative, explained above. Let usstress that this definition does not appeal to knot diagrams, but directly togenuine knots embedded in R3.

It does not take long to understand that Vassiliev’s extension from Kn−1

to Kn is well defined, i.e. does not depend on the choice of a double pointto resolve. Indeed, the calculation of f(K), K ∈ Kn, is in any case reducedto the complete resolution of the knot K which yields an alternating sum

(2) v(K) =∑

ε1=±1

· · ·∑

εn=±1

(−1)|ε|v(Kε1,...,εn),

where ε = (ε1, ..., εn), |ε| is the number of −1’s in the sequence ε1, ..., εn,and Kε1,...,εn is the knot obtained from K by a positive or negative resolutionof the double points according to the sign of εi for the point number i. Thegeometrical background of this phenomenon is that, in the vicinity of thesingular knot K, the pair

(space of immersions S1 → R3, discriminant)

is diffeomorphic to the pair

(Rn, union of coordinate hyperplanes)

multiplied by a vector subspace of codimension n, as shown above in apicture corresponding to the case n = 2. A locally constant function vdefined in the complement of the coordinate cross in Rn is extended to theorigin by the rule (2) and this number is equal to the difference of the twosimilar combinations of order n− 1 corresponding to the two points on anycoordinate axis in Rn lying on either side of the origin.

3.2. Definition of Vassiliev invariants

3.2.1. Definition. (V. Vassiliev [Va1]). A function v : K → G is a Vassilievinvariant of order ≤ n if its extension vanishes on all singular knots withmore than n double points:

v∣∣Kn+1

= 0

(and hence v∣∣Km

= 0 for all m > n).


Another way to state this definition is as follows. Let ∇ : C(Kn) →C(Kn+1) be the operator of Vassiliev’s extension from the space of functionson singular knots with n double to the space of functions on singular knotswith n+ 1 double points:

∇(f)(K) = f(K+)− f(K−),

where K is a knot with n + 1 double points, K+ and K− are its positiveand negative resolution at one and the same double point (we know thatthe difference does not depend on the choice of this point, although eachsummand does). Now, a function f ∈ C(K0) is a Vassiliev invariant of degree≤ n, if it satisfies the difference equation ∇n+1(f) = 0. Note the obviousanalogy between Vassiliev invariants as a subspace of all knot invariantsand polynomials as a subspace of smooth functions on a real line: the roleof differentiation is played by the operator ∇. In these terms the mainopen problem of the theory of finite type invariants (do they constitute acomplete set of knot invariants) is similar to the problem of approximationof functions by polynomials (positive solution is well known).

3.2.2. Example. ([BN0]). The n-th coefficient of the Conway polynomialis a Vassiliev invariant of order ≤ n.

Indeed, the definition of the Conway polynomial, together with Vas-

siliev’s skein relation implies that C( ) = tC( ). Applying this

relation several times, we get

C( . . . ) = tkC( . . . )

for a singular knot with k double points. If k ≥ n + 1, then the coefficientcn of tn in this polynomial is 0.

Notation: Vn will stand for the set of Vassiliev invariants of order ≤ nwith values in the group under consideration. If necessary, we can denotethis group explicitly, using the notation like VQ

n , VZn , etc. If G is a ring, then

the set VGn is a module over G.

3.2.3. Proposition. V0 = const, dimV0 = 1.

Proof. Let f ∈ V0. By definition, the value of (the prolongation of) f onany singular knot with one double point is 0. Pick an arbitrary knot K.Any diagram of K can be turned into a diagram of the trivial knot K0 bycrossing changes (undercrossing ↔ overcrossing) done one at a time. In thelanguage of the functional space, this means that we can connect K withK0 by a path in the space of immersions which intersects the discriminant

3.2. Definition of Vassiliev invariants 61

only at a finite number of generic points. By assumption, the jump of f atevery such point is 0, therefore, f(K) = f(K0). Thus f = const.

3.2.4. Proposition. V1 = const, dimV1 = 1.

Proof. A similar argument shows that the value of v on any knot with 1double point is equal to its value on the special figure ‘infinity’ singular knotand hence to 0:

v( ) = v( ) = 0

Therefore, V1 = V0.

At this point, one is tempted to conclude that dimVn = 1 for any n. Ofcourse, this cannot be true — otherwise there would be no theory of finitetype invariants. In fact, when n = 2 we get the first nontrivial Vassiliev in-variant, namely, the second coefficient c2 of the Conway polynomial. Beforegiving its definition, let us explain the reason why the argument like abovefails in this case. Take a knot with two double points and try to transformit to a fixed knot with two double points using a smooth homotopy whereadditional double points are allowed. It is easy to understand that any knotwith two double points can be reduced to one of the following two canonicalknots:

Special knot K1 Special knot K2

— but these two knots are not equivalent to each other! The essentialdifference between them is in the order of the double points on the curve.

When traveling along the first knot, K1, the two double points are en-countered in the order 1122 (or 1221, 2211, 2112 if you start from a differentinitial point). For the knot K2 the sequence is 1212 (equivalent to 2121).The two sequences 1122 and 1212 are different even if cyclic permutationsare allowed.

Now take an arbitrary singular knot K with two double points. If thecyclic order of these points is 1122, then we can transform the knot to K1,passing in the process of deformation through some singular knots with threedouble points; if the order is 1212, we can reduce K in the same way to thesecond canonical knot K2.

The above argument shows that, to any F-valued order 2 Vassiliev in-variant there corresponds a function on the set of two elements K1,K2with values in F. We thus obtain a linear mapping V2 → F2. The kernel


of this map is equal to V1: indeed, the fact that a given invariant f ∈ V2

satisfies f(K1) = f(K2) = 0 means that it vanished on any singular knotwith 2 double points, which is by definition equivalent to the fact f ∈ V1.

This proves that dimV2 ≤ 2. In fact, dimV2 = 2, because, the invari-ant c2 (the second coefficient of Conway’s polynomial) is different from aconstant (see Table 2.3.5).

3.3. The Casson invariant

The second coefficient of Conway’s polynomial is the simplest non-trivialVassiliev invariant. It has order 2 and is also called the Casson invariant.It can be computed by any plane diagram of the knot according to thefollowing procedure.

Theorem. For any plane diagram of a knot K with an arbitrary base pointwe have:

c2(K) =∑

i j i jU OOU

εiεj ,

where i, j are numbers of two crossing points, εi, εj are local writhe numbers(see p. 19) at these points, and the sum is taken over all pairs of crossingpoints i, j such that, when moving along the knot in the positive direc-tion starting from the base point, these points appear in the specified or-der (i, j, i, j) and the corresponding consecutive passages are undercrossing,overcrossing, overcrossing and undercrossing, respectively.

Proof. Let us prove that the given formula defines a Vassiliev invariant ofdegree 2 taking value 0 on the unknot and value 1 on the left trefoil. Sincethe same holds for the invariant c2 and dimV2 = 2, the assertion of thetheorem will follow.

We must verify three things:

(1) The value of a2(K) does not depend on the choice of the base pointon G.

(2) The value of a2(K) does not depend on the choice of a plane dia-gram of the knot (i. e. it is preserved by Reidemeister moves).

(3) It is a Vassiliev invariant of order 2.

Let us do that step by step.

(1) It is enough ot prove that the base point can be moved one position(i.e. jump over the endpoint of one chord) without changing the value ofthe expression a2(G). Compare the sums for the same knot with the basepoints b1, b2 situated on either side of a certain endpoint e of a chord c:

3.3. The Casson invariant 63

− ++

++

5

2

3

4

1 5

4

3

2

c

exb1

− ++

++

5

2

3

4

1 5

4

3

2

c

eb1

xb

2

If the endpoint e corresponds to an underpass, then the sums in a2(G1)and a2(G2) are identical. Otherwise, the difference a2(G2)−a2(G1) involvesonly the pairs of chords that include the distinguished chord c. In fact,the definition of a2(G) implies that this difference is equal to the total flowthrough the chord c in G. By the flow, we understand the sum of flows of allchords that intersect c, while an individual flow is equal to the sign of thechord and having the same direction. In the example shown in the picture,there are 3 chords that intersect c going from left to right; their signs sumup to 1 + 1 − 1 = 1. At the same time, there is one chord that goes fromright to left carrying a flow equal to 1. Therefore, the total flow of all chordsis 0 in our case.

To prove that the total flow is always 0, let us change the knot in avicinity of the crossing point that corresponds to the chord c taking care ofthe orientation; we will get a two-component link:

A

B

+

−

+

+

1

4

2

3

5*

and it is easy to understand that the flow in either direction is equal to thelinking number of the two components A and B (see section 2.2) computedin two ways. Since the linking number is invariant, the difference is 0.

(2) The fact that a2(G) is preserved under Reidemeister moves, is evidentfor the move V Ω1, because a short chord cannot participate in the right-handpart of the formula. For the moves of second type, we use property (1) andput the base point anywhere on the dotted part of the circle (not between thetwo distinguished chords c1 and c2); then the contribution to the sum of anypair that contains the chord c1 cancels with the corresponding contributionfor c2, and the result is 0. Finally, for type 3 moves, the equivalent diagramsdiffer only by interchanges of adjacent endpoints of distinguished chords,


therefore for a basepoint which is far apart, the sums in a2(G) are identical.

(3) This fact directly follows from Proposition 13.3.4.

Remark. This was actually an example of a Gauss diagram formula, seeChapter 13 for a general construction.

3.4. Chord diagrams

Now let us give a formal definition of the combinatorial structure whichunderlies the above analysis.

Definition. A chord diagram of order n (or degree n) is an oriented circlewith a distinguished set of n pairs of distinct points, considered up to anorientation preserving diffeomorphism of the circle. The set of all chorddiagrams of order n will be denoted by CDn.

Usually, we will not indicate the orientation of the circle on chord dia-grams, assuming that it is oriented counterclockwise.

Examples.

CD1 = ,

CD2 = , ,

CD3 = , , , , .

Remark. Chord diagrams that differ by a reflection with respect to a di-ameter are, in general, different:

6=

This observation reflects the fact that we are studying oriented knots.

3.4.1. Combinatorial definition of chord diagrams. To describe achord diagram in pure combinatorial terms, we can assign labels to theendpoints of chords, equal for the same chord and different for differentchords:

3.4. Chord diagrams 65

a

cd

ac d

b b

and then read the labels off the circle according to the orientation (coun-terclockwise): abcdbacd. The word obtained depends on the choice of thelabels and the starting point. Bringing up the cooresponding equivalencerelation, we arrive at the following equality: CDn = Wn

2 /(Sn × Z2n), whereWn

2 is the set of words of length 2n in n letters each of which appears twice,Sn is the group of permutations (renamings) of these n letters, and Z2n isthe group of cyclic shifts of the word.

3.4.2. Chord diagram of a singular knot. Chord diagrams are used tostore certain information about singular knots.

Definition. The chord diagram σ(K) ∈ CDn of a singular knot K ∈ Kn isthe parameterizing circle S1 equipped with n pairs of points which are theinverse images of n double points of the knot.

Examples.

σ( ) = ,

σ( ) = .

3.4.3. Proposition. (V. Vassiliev [Va1]). The value of v ∈ Vn (Vassilievinvariant of order ≤ n) on a knot K ∈ Kn (knot with n simple double points)depends only on σ(K) (the chord diagram of K):

σ(K1) = σ(K2)⇒ v(K1) = v(K2).

In other words, the mapping Kn → G defined by an G-valued Vassilievinvariant of order ≤ n, factors through the canonical mapping of passing tochord diagrams σ : Kn → CDn:


Kn

CDn

Gσ

∀

∃!?

HHHHHHj

*

Proof. Suppose that σ(K1) = σ(K2). Then there is a one-to-one corre-spondence between the double points of K1 and K2 which is consistent withthe two chord diagrams. Put K1,K2 in R3 so that the corresponding doublepoints coincide together with both branches of the knot in the vicinity ofeach double point.

Knot K1 Knot K2

Now we can find a smooth isotopy from K1 to K2, identical in theneighborhoods of their double points and entirely consisting of knots withn or n + 1 double points. By Vassiliev’s skein relation, this implies thatv(K1) = v(K2).

Proposition 3.4.3 shows that there is a well defined mapping αn : Vn →CD∗n (the group of G-valued functions on the set CDn):

αn(v)(D) = v(K),

where K is an arbitrary knot with σ(K) = D.

To understand the size and the structure of the space Vn, it is very usefulto have a description of the kernel and the image of this mapping αn.

The answer to the first question follows immediately from the definitions:kerαn = Vn−1. Therefore, we obtain an injective homomorphism

(1) αn : Vn/Vn−1 → CD∗n.

Since the mapping αn discards the order (n − 1) part of a Vassilievinvariant v, we can, by analogy with differential operators, call the functionαn(v) on chord diagrams the symbol of the Vassiliev invariant v:

symb(v) = αn(v), n = ord(v).

3.5. Goussarov’s approach 67

3.4.4. Corollary. If the group G is a field, then the set of G-valued Vassilievinvariants of degree no greater than n, is a finite-dimensional vector spaceover G.

A complete description of the image, imαn, which is more involved, willbe given in Theorem 4.2.1, p. 84.

3.5. Goussarov’s approach

Vassiliev’s definition of finite type invariants relies on the extension of in-variants to singular knots which is only possible when a coorientation of thediscriminant Σ is chosen.

One can, however, give an equivalent definition which does not referto the positive and negative resolutions of double points on the singularknot. Let K be a knot given by a plane diagram D with no less than ncrossings out of which exactly n are distinguished and numbered. Then forany sequence ε = (ε1, . . . , εn) of signs εi = ±1 one can define a new knotKε represented by the diagram obtained from the diagram D by crossingchanges at all points number i with εi = −1.

3.5.1. Definition. (M. Goussarov). A knot invariant f is said to be a finitetype invariant of degree at most n− 1 if for any knot K given by a diagramas described above we have

(1)∑

ε

(−1)|ε|f(Kε1,...,εn) = 0,

where ε = (ε1, ..., εn), |ε| is the number of −1’s in the sequence ε1, ..., εn,and the summation is spread over all the 2n choices of signs.

Here is an example of such a sum:

f( )

− f( )

− f( )

+ f( )

The fact that this definition is equivalent to Vassiliev’s one can be seenas follows. Take a singular knot K with n double points, choose an arbitraryplane diagram of K and, using Vassiliev’s skein relation, expand it into analternating sum of 2n knots where all these points are resolved. This sum,possibly after multiplication by −1, has the same appearance as the left-hand side of Equation (1).


3.5.2. Goussarov’s filtration. Linear combinations of knots appearing inEquation 1 lead to a filtration in the space of knots X (see Section 1.7). LetXn be the subspace of X spanned by all elements

(2)∑

ε

(−1)|ε|f(Kε1,...,εn).

For n = 0 we set by definition X0 = X .

The subspaces Xn form a decreasing filtration (see Section A.2.13) in X ,because a combination 2 with a given n can be represented as a differenceof two such combinations of order n− 1 and thus Xn ⊂ Xn−1.

Although it is not immediately clear, the quotient spaces Xn/Xn−1 ofGoussarov’s filtration are finite-dimensional. As an illustration, let us provethat dimX/X1 = 1. First, it is evident that X1 6= X , because every elementof X1 has a zero sum of coefficients. Secondly, according to Exercise 7 everyknot can be turned into the unknot after the addition of several combinationsof the form K+ −K− and hence dimX/X1 ≤ 1.

For an arbitrary n, it is not easy to prove that dimXn/Xn−1 < ∞,staying entirely within Goussarov’s approach. This fact follows from theanalysis of the space of chord diagrams related to singular knots (see Chapter4). In general, although the two theories — Vassiliev’s and Goussarov’s —are basically equivalent, Vassiliev’s definition has the advantage of beingmore constructive.

3.6. Algebra of Vassiliev invariants

The set of all Vassiliev invariants forms a graded commutative algebra withrespect to the usual (pointwise) multiplication of functions.

3.6.1. Theorem. VmVm ⊂ Vm+n, i.e. if f and g are Vassiliev invariantsof degrees no greater than m and n, respectively, then fg is a Vassilievinvariant of degree no greater than m+ n.

Example. For m = n = 1 the fact that we must prove reads: if twoknot invariants have the property

f(L++)− f(L+−)− f(L−+) + f(L−−) = 0,

g(L++)− g(L+−)− g(L−+) + g(L−−) = 0

3.7. Classical knot polynomials as Vassiliev invariants 69

for any singular knot L with 2 double points, then their product satisfiesthe equation

f(K+++)g(K+++)− f(K++−)g(K++−)

− f(K+−+)g(K+−+) + f(K+−−)g(K+−−)

− f(K−++)g(K−++) + f(K−+−)g(K−+−)

+ f(K−−+)g(K−−+)− f(K−−−)g(K−−−) = 0

for any singular knot K with 3 double points. Of course, we already knowthat first order Vassiliev invariants are only constants, and on top of thisknowledge the previous fact is obvious; however, it is not so obvious forarbitrary m and n, and this example gives a feeling of the kind of algebraicdifficulties one has to overcome.

Proof. Following V. Arnold [Ar2], let us notice that for any two knot in-variants f and g we have

∇(fg) = f+g+−f−g− = f+g+−f−g++f−g+−f−g− = ∇(f)g++f−∇(g),

where ∇ is the difference operator defined on page 60. Note that g+ and f−

refer to the positive and negative resolutions of one and the same point.

The previous equation shows that ∇ satisfies kind of a Leibnitz rule,therefore its (m + n + 1)-th power, evaluated at the the product fg, willcontain, in each term of its full expansion, either ∇k(f) with k ≥ m+ 1 or∇k(g) with k ≥ n + 1. Therefore ∇m+n+1(fg) = 0, if f is an invariant ofdegree m and g an invariant of degree n.

The situation, however, is not as simple as it looks. In fact, any ex-pansion of ∇p(fg) involves not only pure powers like ∇k(f) or ∇k(g), butalso terms like ∇(∇(f+)−), and one must keep track of the double pointsto which the signs in the superscripts refer. This can be done, but we prefernot to go into detail here, postponing a complete proof of this theorem untilSection 4.4 (Proposition 4.4.4) where it will follow from the properties ofthe Goussarov filtration and general facts about filtered bialgebras.

3.7. Classical knot polynomials as Vassiliev invariants

In Example 3.2.2 we have seen that the coefficients of the Conway polynomialare Vassiliev invariants. The Conway polynomial, taken as a whole, is not, ofcourse, a finite type invariant, but it is an infinite linear combination of such.The important thing is that two knots K1 and K2 have the same Conwaypolynomial if v(K1) = v(K2) for any Vassiliev invariant v. This propertyholds for all classical knot polynomials — only the ‘infinite combination’ inquestion does not need to be linear. For example, the Jones polynomial is


an ‘infinite nonlinear combination’ of Vassiliev invariants in the followingexact sense.

3.7.1. Substitute t = eh in the Jones polynomial (Sec. 2.4) of a knot K andexpand J(K) into a formal series over the powers of h. Let jn(K) be thecoefficient of hn in this expansion.

Theorem. ([BL, BN1]) The coefficient jn(K) is a Vassiliev invariant oforder ≤ n.

Proof. Plugging t = eh = 1 +h+ . . . into the skein relation from Sec. 2.4.4we get

(1− h+ . . . ) · J( )− (1 + h+ . . . )J( ) = (h+ . . . )J( )

From this we can conclude that that the difference J( ) = J( )−

J( ) is congruent to 0 modulo h. Therefore for a singular knot Kk with

k double points the Jones polynomial J(Kk) is divisible by hk. In particular,for k ≥ n+ 1 the coefficient of hn equals zero.

Now we will give an explicit description of symbols of the finite typeinvariants jn; the similar problem for the Conway polynomial is left as anexercise (15 at the end of the chapter, page 77).

3.7.2. Symbol of the Jones invariant jn(K). To find the symbol ofjn(K), we must compute the coefficient of hn in the Jones polynomial J(Kn)of a singular knot Kn with n double points in terms of its chord diagramσ(Kn). Since

J( )= J( )−J( )=h(j0( )+j0( )+j0( )

)+. . .

the contribution of a double point of Kn to the coefficient jn(Kn) equalsthe sum of values of j0(·) on the three links in the parentheses above. Thevalues of j0(·) for the last two links are equal because, according to exercise

4 of this chapter, j0(L) = (−2)#(components of L)−1. So it does not de-pend on the specific linkage and knottedness of L and we can freely changethe under/over-crossings of L. On the level of chord diagrams these twoterms mean that we just forget about the chord corresponding to this dou-

ble point. The first term, j0( ), corresponds to the smoothing of the


double point according to the orientation of our knot (link). On the level ofchord diagrams this corresponds to the doubling of a chord:

.

This leads to the following procedure of computing the value of the symbolof jn(D) on a chord diagram D. Introduce a state s for D as an arbitraryfunction on the set chords of D with values in the set 1, 2. With eachstate s we associate an immersed plane curve obtained from D by resolutions(either doubling or deletion) of all its chords according to s:

c , if s(c) = 1; c , if s(c) = 2.

Let |s| denote the number of components of the curve obtained in this way.Then

symb(jn)(D) =∑

s

(∏

c

s(c))

(−2)|s|−1 ,

where the product is taken over all n chords of D, and the sum is takenover all 2n states for D.

For example, to compute the value of the symbol of j3 on the chord

diagram we must consider 8 states:

Qs(c)=1

|s|=2

Qs(c)=2

|s|=1

Qs(c)=2

|s|=1

Qs(c)=2

|s|=3

Qs(c)=4

|s|=2

Qs(c)=4

|s|=2

Qs(c)=4

|s|=2

Qs(c)=8

|s|=1

Therefore,

symb(j3)( )

= −2 + 2 + 2 + 2(−2)2 + 4(−2) + 4(−2) + 4(−2) + 8 = −6


Similarly one can compute the values of symb(j3) on all chord diagrams withthree chords. Here is the result.

D

symb(j3)(D) 0 0 0 −6 −12

This function on chord diagrams, as well as the whole Jones polynomial,is closely related to the Lie algebra sl2 and its standard 2-dimensional re-presentation. We will return to this subject several times in the sequel (seeSections 7.1.3, 7.1.6 etc).

3.7.3. According to exercise 8 (p. 53) for the mirror reflection K of a knotK the power series expansion of J(L) can be obtained from the series J(K)by substituting −h for h. This means that j2k(K) = j2k(K) and j2k+1(K) =−j2k+1(K).

3.7.4. Table 1 displays the first five terms of the power series expansion ofthe Jones polynomial after the substitution t = eh.

31 1 −3h2 +6h3 −294 h

4 +132 h

5 +. . .

41 1 +3h2 +54h

4 +. . .

51 1 −9h2 +30h3 −2434 h4 +185

2 h5 +. . .

52 1 −6h2 +18h3 −652 h

4 +872 h

5 +. . .

61 1 +6h2 −6h3 +172 h

4 −132 h

5 +. . .

62 1 +3h2 −6h3 +414 h

4 −252 h

5 +. . .

63 1 −3h2 −174 h

4 +. . .

71 1 −18h2 +84h3 −4772 h4 +511h5 +. . .

72 1 −9h2 +36h3 −3514 h4 +159h5 +. . .

73 1 −15h2 −66h3 −6974 h4 −683

2 h5 +. . .

74 1 −12h2 −48h3 −113h4 −196h5 +. . .

75 1 −12h2 +48h3 −119h4 +226h5 +. . .

76 1 −3h2 +12h3 −894 h

4 +31h5 +. . .

77 1 +3h2 +6h3 +174 h

4 +132 h

5 +. . .

81 1 +9h2 −18h3 +1354 h4 −87

2 h5 +. . .

82 1 −6h3 +27h4 −1332 h5 +. . .

83 1 +12h2 +17h4 +. . .


84 1 +9h2 −6h3 +634 h

4 −252 h

5 +. . .

85 1 +3h2 +18h3 +2094 h4 +207

2 h5 +. . .

86 1 +6h2 −18h3 +772 h

4 −1232 h5 +. . .

87 1 −6h2 −12h3 −472 h

4 −31h5 +. . .

88 1 −6h2 −6h3 −292 h

4 −252 h

5 +. . .

89 1 +6h2 +232 h

4 +. . .

810 1 −9h2 −18h3 −1234 h4 −75

2 h5 +. . .

811 1 +3h2 −12h3 +1254 h4 −55h5 +. . .

812 1 +9h2 +514 h

4 +. . .

813 1 −3h2 −6h3 −534 h

4 −252 h

5 +. . .

814 1 +6h4 −18h5 +. . .

815 1 −12h2 +42h3 −80h4 +1872 h5 +. . .

816 1 −3h2 +6h3 −534 h

4 +372 h

5 +. . .

817 1 +3h2 +294 h

4 +. . .

818 1 −3h2 +74h

4 +. . .

819 1 −15h2 −60h3 −5654 h4 −245h5 +. . .

820 1 −6h2 +12h3 −352 h

4 +19h5 +. . .

821 1 −6h3 +21h4 −852 h

5 +. . .

Table 1: Taylor expansion of modified Jones’ polynomial

3.7.5. Example. In the following examples the h-expansion of the Jonespolynomial starts from a power of h equal to the number of double pointsin a singular knot, in compliance with Theorem 3.7.1.

J

( )= J

( )

︸︷︷︸||0

−J( )

= − J( )

︸︷︷︸||1

+J

( )

= −1 + J(31) = −3h2 + 6h3 − 294 h

4 + 132 h

5 + . . .


J

( )= J

( )− J

( )

︸︷︷︸||0

= J

( )− J

( )

︸︷︷︸||1

= J(31)− 1 = −3h2 − 6h3 − 294 h

4 − 132 h

5 + . . .

J

( )= J

( )− J

( )= −12h3 − 13h5 + . . .

3.7.6. J. Birman and X.-S. Lin proved in [Bir2, BL] that all quantuminvariants have the property proved above for the Jones polynomial. Moreprecisely, let θ(K) be the quantum invariant constructed as in Sec. 2.6. Itis a polynomial in q and q−1. Now let us make a substitution q = eh andconsider the coefficient θn(K) of hn in the Taylor expansion of θ(K).

Theorem. ([BL, BN1]) The coefficient θn(K) is a Vassiliev invariant oforder ≤ n.

Proof. The argument is similar to that used in Theorem 3.7.1: it comesfrom the fact that an R-matrix R and its inverse R−1 are congruent moduloh.

3.8. Actuality tables

Actuality tables (term proposed by V. Vassiliev) provide a way to organizethe data necessary for the computation of a single finite type invariant.

3.8.1. Proposition 3.4.3 states the coincidence of values of a Vassiliev in-variant v of order ≤ n on any two singular knots K1 and K2 with precisely ndouble points each and with the same chord diagram σ(K1) = σ(K2). Nowsuppose that K1 and K2 are singular knots with k double points and k isless than n. How are the values v(K1) and v(K2) related to each other?

In the proof of Proposition 3.4.3 we saw that K1 can be obtained fromK2 by a finite number of crossing changes. According to Vassiliev’s skeinrelation (1), each crossing change means that the difference of values of v onthe two knots equals to the value of v on a knot with an additional doublepoint. So, in our case, the difference v(K1) − v(K2) can be expressed as acombination of values of v on certain knots with k + 1 double points.

Therefore, if we know the values of v on all knots with ≥ k + 1 doublepoints and on one particular knot for each chord diagram with k double

3.8. Actuality tables 75

points, then we can compute the value of v on an arbitrary knot with kdouble point. This leads us to the data sufficient to completely determine asingle Vassiliev invariant. Following Vassiliev [Va1] and Birman, Lin [BL,Bir2], we collect this data in the form of an actuality table.

3.8.2. To construct the actuality table we must choose a representative(canonical) singular knot for every chord diagram. This can be done ina variety of ways, but of course, it is more convenient to choose not toocomplicated knots. For example, for n = 0 we choose the unknot, for n = 1, 2we use the canonical knots chosen in Sec. 3.2. A possible choice of canonicalknots up to degree 3 is shown in the table.

CD0 CD1 CD2 CD3

The actuality table for a particular invariant v of order ≤ n consists of itssymbol and the set of its values on the a chosen set of canonical knots withup to n double points. By Proposition 3.4.3, the values of v on the knotswith precisely n double points depend only on their chord diagrams, but theprevious values in the actuality table depend not only on chord diagramsbut also on the representing canonical knots. Of course, the values in theactuality table cannot be arbitrary. They satisfy certain relations which weshall discuss later (see Sec. 4.1).

3.8.3. Example. The second coefficient c2 of the Conway polynomial(Sec. 3.2) is a Vassiliev invariant of order ≤ 2. Here is an actuality table forit.

c2 : 0 0 0 1

Note that the order of values in the table corresponds to the order of canon-ical knots in the table of representatives.

3.8.4. Example. A Vassiliev invariant of order 3 is given by the thirdcoefficient j3 of the Taylor expansion of Jones polynomial (Sec. 2.4). Theactuality table for j3 looks as follows.

j3 : 0 0 0 6 0 0 0 −6 −12


3.8.5. To illustrate the general procedure of computing the value of a Vas-siliev invariant on a particular knot by means of actuality tables let uscompute the value of j3 on the right-hand trefoil. The right-hand trefoil isan ordinary knot, without singular points, so we have to deform it (usingcrossing changes) to our canonical knot without double points, i. e. theunknot. This can be done by one crossing change, and by Vassiliev’s skeinrelation (Eq. 1 on page 59) we have

j3

( )= j3

( )+ j3

( )= j3

( )

because j3(unknot) = 0 in the actuality table. Now the knot with one doublepoint we got is not quite the one from our canonical knots. We can deformit to our canonical knot changing the upper right crossing.

j3

( )= j3

( )+ j3

( )= j3

( )

Here we used the fact from the actuality table that j3 vanishes on the canon-ical knot with a single double point. The knot with two double points on theright-hand side of the equation still differs by one crossing from the canoni-cal knot with two double points. This means that we have to do one morecrossing change. Combining these equations together and using the valuesfrom the actuality table we get the final answer

j3

( )= j3

( )= j3

( )+ j3

( )= 6− 12 = −6

3.8.6. The first ten Vassiliev invariants. Using actuality tables, onecan find the values of basic Vassiliev invariants of small degree. Table 2displays a certain basis in the space of Vassiliev invariants up to degree 5. Itrepresents an abridged version of the table compiled by T. Stanford [Sta1],where the values of basic invariants up to degree 6 are given on all knots upto 10 crossings.

Some of the entries in Table 2 are different from [Sta1], this is due to thefact that, for some non-amphicheiral knots, Stanford uses mirror reflectionsinstead of the canonical Rolfsen’s knots depicted in Table 1.

The two signs after the knot number refer to their symmetry peoperties:a plus in the first position means that the knot is amphicheiral, a plus inthe second position means that the knot is invertible.

3.9. The case of framed knots TBW

!!! — will be written; in particular, we will prove that the self-linking numberdefined in Section 2.2.3 is a Vassiliev invariant of order 1 taking value 1 onthe chord diagram with one chord.

Exercises 77

Exercises

(1) Using actualiy tables, compute the value of j3 on the left-hand trefoil.

(2) Choose the canonical knots with four double points and construct theactuality tables for the forth coefficients c4 and j4 of the Conway andJones polynomials.

(3) Prove that j0(K) = 1 and j1(K) = 0 for any knot K.

(4) Show that the value of j0 on a link with k components is equal to(−2)k−1.

(5) Prove that for any knot K the integer j3(K) is divisible by 6.

(6) For a knot K, find the relation between the second coefficients c2(K)and j2(K) of the Conway and Jones polynomials.

(7) Let T be the trefoil knot 31 and let K be the granny knot 31#31. Provethat v(K) = 2v(T )− v(1) for any Vassiliev invariant v ∈ V3.

(8) Prove that for a knot K the n-th derivative at 1 of the Jones polynomial

dn(J(K))

dtn

∣∣∣∣t=1

is a Vassiliev invariant of order ≤ n. Use the skein relation to find thefirst derivative. Find the relation between these invariants and j1, . . . , jnfor small values of n.

(9) Express the coefficients c2, c4, j2, j3, j4, j5 of the Conway and Jonespolynomials in terms of the basic Vassiliev invariants from Table 2.

(10) Find the symbols of several table Vassiliev invariants (Table 2).

(11) Express the invariants of Table 2 through the coefficients of the Conwayand the Jones polynomials.

(12) Find the actuality tables for some of the Vassiliev invariants appearingin Table 2.

(13) Explain the correlation between the first sign and the zeroes in the lastfour columns of Table 2.

(14) Check that Vassiliev invariants up to order 4 are enough to distinguish,up to orientation, all table knots with no more than 8 crossings.

(15) Prove that the symbol of the coefficient cn of the Conway polynomial canbe calculated as follows. Double every chord of a given chord diagramD like in Sec. 3.7.2, and let |D| be equal to the number of components


of the obtained curve. Then

symb(cn)(D) =

1, if |D| = 10, otherwise .

(16) Prove that there is a well-defined prolongation of knot invariants tosingular knots with a non-degenerate triple point according to the rule

f( ) = f( )− f( ) .

Is it true that, according to this prolongation, a Vassiliev invariant ofdegree 2 is equal to 0 on any knot with a triple point?

Is it possible to use the same method to define a prolongation of knotinvariants to knots with self-intersections of multiplicity higher than 3.

(17) Following Example 3.7.5, find the power series expansion of the modified

Jones polynomial of the singular knot .

(18) Prove the following relation between the Casson knot invariant c2, ex-tended to singular knots, and the linking number of two curves. LetK be a knot with one double point. Smoothing the double point by

the rule 7→ , one obtains a 2-component link L. Then

lk(L) = c2(K).

Exercises 79

v0 v2 v3 v41 v42 v22 v51 v52 v53 v2v3

31 −+ 1 1 −1 1 −3 1 −3 1 −2 −1

41 ++ 1 −1 0 −2 3 1 0 0 0 0

51 −+ 1 3 −5 1 −6 9 −12 4 −8 −15

52 −+ 1 2 −3 1 −5 4 −7 3 −5 −6

61 −+ 1 −2 1 −5 5 4 4 −1 2 −2

62 −+ 1 −1 1 −3 1 1 3 −1 1 −1

63 ++ 1 1 0 2 −2 1 0 0 0 0

71 −+ 1 6 −14 −4 −3 36 −21 7 −14 −84

72 −+ 1 3 −6 0 −5 9 −9 6 −7 −18

73 −+ 1 5 11 −3 −6 25 16 −8 13 55

74 −+ 1 4 8 −2 −8 16 10 −8 10 32

75 −+ 1 4 −8 0 −5 16 −14 6 −9 −32

76 −+ 1 1 −2 0 −3 1 −2 3 −2 −2

77 −+ 1 −1 −1 −1 4 1 0 2 0 1

81 −+ 1 −3 3 −9 5 9 12 −3 5 −9

82 −+ 1 0 1 −3 −6 0 2 0 −3 0

83 ++ 1 −4 0 −14 8 16 0 0 0 0

84 −+ 1 −3 1 −11 4 9 0 −2 −1 −3

85 −+ 1 −1 −3 −5 −5 1 −5 3 2 3

86 −+ 1 −2 3 −7 0 4 9 −3 2 −6

87 −+ 1 2 2 4 −2 4 7 −1 3 4

88 −+ 1 2 1 3 −4 4 2 −1 1 2

89 ++ 1 −2 0 −8 1 4 0 0 0 0

810 −+ 1 3 3 3 −6 9 5 −3 3 9

811 −+ 1 −1 2 −4 −2 1 8 −1 2 −2

812 ++ 1 −3 0 −8 8 9 0 0 0 0

813 −+ 1 1 1 3 0 1 6 0 3 1

814 −+ 1 0 0 −2 −3 0 −2 0 −3 0

815 −+ 1 4 −7 1 −7 16 −16 5 −10 −28

816 −+ 1 1 −1 3 0 1 2 2 2 −1

817 +− 1 −1 0 −4 0 1 0 0 0 0

818 ++ 1 1 0 0 −5 1 0 0 0 0

819 −+ 1 5 10 0 −5 25 18 −6 10 50

820 −+ 1 2 −2 2 −5 4 −1 3 −1 −4

821 −+ 1 0 1 −1 −3 0 1 −1 −1 0

Table 2. Basic Vassiliev invariants of order ≤ 5

Chapter 4

Chord diagrams

The definition of finite type invariants given in the previous chapter led usnaturally to the notion of a chord diagram. Functions on the set of chorddiagrams of a certain order can be considered as leading part of finite typeinvariants of the corresponding degree. Studying the behaviour of (Vassilievextended) invariants in a neighborhood of a knot with a triple point, we willfind that these functions satisfy two sets of relations: more important four-term relations and less important one-term relations. It is very importantthat the vector space generated by chord diagrams modulo these relations,has a rich algebraic structure, namely that of a Hopf algebra (see Appendixat the end of the book). This structure is of primary concern in the presentchapter.

4.1. Four- and one-term relations

Assume, from now on, that F is a fixed field and Vn denotes the space ofF-valued Vassiliev invariants of order ≤ n. We will often suppose that thecharasteristic of F is 0 and use the field of complex numbers C if we needto be specific. In Section 3.2 (Eq. (1) on page 66) we constructed a linearinclusion

αn : Vn/Vn−1 → CD∗n,

where CD∗n is the vector space of F-valued functions on the set CDn of chorddiagrams of order n.

To describe the image of αn, we will need the following definitions.

4.1.1. Definition. A weight system of order n is a function f ∈ CD∗n thatsatisfies 4-term relations.

81

82 4. Chord diagrams

4.1.2. Definition. A 4-term relation says that the alternating sum of thevalues of f on certain special quadruples of diagrams is zero:

(1) f( )− f( ) + f( )− f( ) = 0.

Here, the dotted arcs suggest that there might be further chords attachedto their points, while on the solid portions of the circle all endpoints of thechords are explicitly shown. For example, this means that in the 1st and2nd diagrams the two bottom points are adjacent. The chords that are notshown must be the same in all the four cases.

Example. Let us find all 4-term relations for chord diagrams of order 3.We must add one chord in one and the same way to all the four terms ofEquation 1. Since there are 3 dotted arcs, there are 6 different ways to dothat, in particular,

f( )− f( ) + f( )− f( ) = 0

and

f( )− f( ) + f( )− f( ) = 0.

Some of the diagrams in these equations are equal, and the relations can be

simplified as = and − 2 + = 0. The reader

is invited to check that the remaining four 4-term relations (we wrote only2 out of 6) are either trivial or coincide with one of these two.

It is often useful to look at the 4T relation from the following viewpoint.Imagine that of the two chords that participate in equation (1) one (“sec-ond”) is fixed, as well as one of the two endpoints of the first one, whileanother endpoint of that first chord makes a complete turn around the firstchord, stopping at four places, adjacent to its endpoints; the resulting fourdiagrams are then summed up with alternating signs. Pictorially:

(2) f( )− f( ) + f( )− f( ) = 0.

where the fixed endpoint of the first (moving) chord is marked by .

4.1. Four- and one-term relations 83

Another way of writing the 4T relation, which will be useful in Section5.1, is to split the four terms in two pairs:

f( )− f( ) = f( )− f( ).

Due to obvious symmetry, this can be completed as follows:

(3) f( )− f( ) = f( )− f( ).

Note that for every order n there are as many 1-term relations as thereare chord diagrams with an isolated chord and the choice of a specific 4-termrelation depends on the following data:

• a diagram of order n− 1,

• a distinguished chord of this diagram (“fixed chord”), and

• a distinguished arc on the circle of this diagram (where the fixedendpoint is placed).

We will see in the next section that the four-term relations are alwayssatisfied by the symbols of Vassiliev invariants, both in the conventional andin the framed case. For the framed knots, there are no other relations; in theunframed case, there is another set of relations, called one-term, or framingindependence relations.

4.1.3. Definition. An unframed weight system of order n is a weight systemf ∈ CD∗n that also satisfies 1-term relations.

4.1.4. Definition. A 1-term relation says that f vanishes on every chorddiagram with an isolated chord. An isolated chord is a chord that does notintersect any other chord of the diagram.

Example.

f( ) = 0,

4.1.5. Notation. Denote by Wn the subspace of CD∗n consisting of allweight systems of order n and by W ′n the subspace of all unframed weightsystems of order n (those that satisfy not only 4T relations, but also 1Trelations).


4.2. The Fundamental Theorem

In Section 3.4 we constructed an injective mapping αn : Vn/Vn−1 → CD∗n.The fundamental theorem about Vassiliev invariants describes its image.

4.2.1. Theorem. (Vassiliev–Kontsevich) Let Vn be the space of Vassilievknot invariants over C. The mapping αn yields an isomorphism

αn : Vn/Vn−1 →W ′n,that is, the space of unframed weight systems W ′ = ⊕∞n=0W ′n is isomorphicto the graded space associated with the filtered space of Vassiliev invariantsV.

The theorem consists of two parts:

• (V. Vassiliev) The symbol of every Vassiliev invariant is an un-framed weight system.

• (M. Kontsevich) Every unframed weight system is the symbol of acertain Vassiliev invariant.

We will now prove the first (easy) part of the theorem. The second(difficult) part will be proved later (in Section 8.8) using the Kontsevichintegral.

The first part of the theorem consists of two assertions, and we provethem one by one.

4.2.2. First assertion: any function f ∈ CD∗n coming from an invariantv ∈ Vn satisfies 1-term relations.

Proof. Let K ∈ Kn be a singular knot whose chord diagram contains anisolated chord. The double point p that corresponds to the isolated chorddivides the knot into two parts: A and B.

p

A

B1

2

3

p

p

1

2

12

3

3

The fact that the chord is isolated means that A and B do not havecommon double points. They may, however, be knotted between themselvesin the sense that on any knot diagram there are crossing points between thetwo parts. Due to Vassiliev’s skein relation, by changing undercrosses andovercrosses, we can untangle part A from part B thus obtaining a singular

4.2. The Fundamental Theorem 85

knot K ′ with the property that the two parts lie on either side of a plane inR3 passing through the double point p:

A

P

B

Here, the two resolutions of the double point p obviously give equivalentknots, therefore v(K) = v(K ′) = v(K ′+)− v(K ′−) = 0.

4.2.3. Second assertion: any function f ∈ CD∗n coming from an invari-ant v ∈ Vn satisfies 4-term relations.

Proof. We will use the following lemma.

Lemma. (4-term relation for knots). Any Vassiliev invariant satisfies

f( )− f( ) + f( )− f( ) = 0,

Proof. By Vassiliev’s skein relation:

f( ) = f( )− f( ) = a− b

f( ) = f( )− f( ) = c− d

f( ) = f( )− f( ) = c− a

f( ) = f( )− f( ) = d− b

The alternating sum of these expressions is (a− b)− (c− d) + (c− a)−(d− b) = 0, and the lemma is proved.

Now, to prove the 4-term relation for the symbols of Vassiliev invariants,let us choose, for the first diagram D1 in the relation, an arbitrary singularknot K1 such that σ(K1) = D1:

σ( ) = .


Then the three remaining knots K2, K3, K4 that participate in the 4-term relation for knots, correspond to the three remaining chord diagramsof the 4-term relation for chord diagrams, and the claim follows from thelemma.

4.2.4. Remark. The 1-term relation is also called framing independencerelation, because there exists a theory of Vassiliev invariants for framedknots, where the 4-term relation holds, whereas the 1-term relation doesnot.

Like in the conventional case, we can introduce the prolongation of in-variants to the singular framed knots, understood as framed knots withsimple double points, and give the definition of finite type invariants. If wedenote the space of such invariants by FVn, then we get a linear mappingFVn/FVn−1 → CD∗n which satisfies the 4T relation for the same reasons,but does not satisfy the 1T relation, because the two knots, differing by akink like in the first Reidemeister move, are equivalent as usual knots, butnot equivalent as framed knots. This shows that FVn/FVn−1 is a subspaceof A∗n. The equality FVn/FVn−1 = A∗n can be proved using the Kontse-vich’s integral for framed knots (see Sec. 10.1 below).

4.2.5. We see that 4T relations are more fundamental than 1T relations,therefore in the sequel we will mainly study combinatorial structures whereonly 4T relations are used. Anyway, 1T relations can always be added,either by considering an appropriate subspace or an appropriate quotientspace (see Section 4.5.5). This is especially easy to do in terms of primitiveelements (see page 97), because the problem is reduced to simply leavingout one primitive generator.

4.3. Intersection graphs

4.3.1. Definition. ([CD1]) The intersection graph Γ(D) of a chord dia-gram D is a graph whose vertices correspond to the chords of D and twovertices are connected by an edge iff the corresponding chords intersect.(Two chords, a and b, are said to intersect if their endpoints a1, a2 and b1,b2 appear in the interlacing order a1, b1, a2, b2 along the circle.)

For example,

4.3. Intersection graphs 87

5

3

1

2

1

5

4

2

3

4 −→1

2

3

4 5

chord −→ vertexintersection −→ edge

Note that not every graph can be obtained as the intersection graph ofa chord diagram.

4.3.2. Exercises. (1) Prove that all graphs with no more than 5 verticesare intersection graphs.(2) Find a graph of degree 6 which is not an intersection graph.

Answer to (2):

Different diagrams may have one and the same intersection graph. Forexample, there are three different diagrams

with the same intersection graph

Intersection graphs contain a good deal of information about chord dia-grams. In [CDL1] the following conjecture was stated.

4.3.3. Intersection graph conjecture. If D1 and D2 are two chord di-agrams whose intersection graphs are equal, Γ(D1) = Γ(D2), then D1 = D2

as elements of A (i.e. modulo four-term relations).

Although wrong in general (see Section 9.4.9), this assertion is true insome particular situations:

1) for all diagrams D1, D2 with n chords up to n = 8 (a direct computercheck);

2) for Vassiliev invariants v coming from standard representations of Liealgebras glN or soN (see Chapter 7) (this means that Γ(D1) = Γ(D2) impliesv(D1) = v(D2));


3) when Γ(D1) = Γ(D2) is a tree (see [CDL2]) or, more generally, D1,D2 belong to the forest subalgebra (see [CDL3]).

4) when Γ(D1) = Γ(G2) is a graph with a single loop (see [Mel1]).

4.3.4. Some weight systems factoring through intersection grapfs.

Chromatic polynomial, etc. TBW

4.4. Bialgebras of knots and knot invariants

Prerequisites on bialgebras can be found in the Appendix (see page 336).

In Section 2.5 we remarked that the algebra of knot invariants I, as avector space, is dual to the space of knots K defined in Section 1.7.

Now, using the construction of dual bialgebras (Section A.2.8), we candefine the coproducts in X and I, using the products in I and X , respec-tively. Explicit formulas are:

δ(K) = K ⊗Kfor a knot K (then extended by linearity to the entire space X ) and

δ(f)(K1 ⊗K2) = f(K1#K2)

for an invariant f and any pair of knots K1 and K2.

4.4.1. Exercise. Define the counits and check the compatibility of theproduct and coproduct in each of the two algebras.

Let us see what are the primitive and group-like elements in the algebrasX and I. As concerns the algebra of knots X , both structures are verypoor: it follows directly from the definitions that P(X ) = 0, G(X ) = 1.(Elements of K ⊂ X , i.e. knots, are semigroup-like, but not group-like!)

The situation is quite different for the algebra of invariants. As a conse-quence of Proposition A.2.12 we obtain a description of primitive and group-like knot invariants (in particular, Vassiliev invariants): this is nothing butadditive and multiplicative invariants, i.e. invariants satisfying the relations

f(K1#K2) = f(K1) + f(K2),

f(K1#K2) = f(K1)f(K2)

respectively for any two knots K1 and K2.

It follows that the sets P(I) and G(I) are quite rich; they are relatedwith each other by the log-exp correspondence.

4.4.2. Theorem. The bialgebra of knots X supplied with the Goussarovfiltration (p. 68) is a d-filtered bialgebra (page 343), i.e. a bialgebra with adecreasing filtration.

4.4. Bialgebras of knots and knot invariants 89

Proof. There are two assertions to prove:

(1) If x ∈ Xm and y ∈ Xn, then xy ∈ Xm+n,

(2) If x ∈ Xn, then δ(x) ∈∑p+q=nXp ⊗Xq.Proof of (1).

It is enough to consider only the generators of Xm and Xn:

x =∑

ε1,...,εm

(−1)|ε|f(Kε1,...,εm), y =∑

ε1,...,εn

(−1)|ε|f(Lε1,...,εn).

Then, evidently,

xy =∑

ε1,...,εm+n

(−1)|ε|f((K#L)ε1,...,εm+n).

Proof of (2).

Here we have to introduce some additional notation.

Let K be a knot given by a plane diagram with ≥ n crossings out ofwhich exactly n are distinguished and numbered. Consider the set K of2n knots that differ from K by crossing changes at the distinguished pointsand the vector space XK (a subspace of the algebra X defined in Sec. 1.7)

spanned by K. The group Zn2 acts on the set K; the action of i-th generatorsi consists in the flip of under/over-crossing at distinguished point numberi. We thus obtain a set of n commuting linear operators si : XK → XK .Set σi = 1 − si. In these terms, a typical generator of Xn can be writtenas x = (σ1 · · · σn)(K). To evaluate δ(x), we must find the commutatorrelations between the operators δ and σi.

4.4.3. Lemma.

δ σi = (σi ⊗ id + si ⊗ σi) δ,where both the left hand side and the right hand side are understood as linearoperators from XK into XK ⊗XK .

Proof. Just check that the values of both operators on an arbitrary elementof the set K are equal.

A successive application of the lemma yields:

δ σ1 · · · σn =n∏

i=1

(σi ⊗ id + si ⊗ σi) δ

= (∑

I⊂1,...,n

∏

i∈Iσi∏

i6∈Isi ⊗

∏

i6∈Iσi) δ .


Therefore, an element x = (σ1 · · · σn)(K) satisfies

δ(x) =∑

I⊂1,...,n(∏

i∈Iσi∏

i6∈Isi)(K)⊗ (

∏

i6∈Iσi)(K)

which obviously belongs to∑

p+q=nXp ⊗Xq.

4.4.4. Proposition. The algebra of knot invariants I is an i-filtered bial-gebra (page 343), i.e. abialgebra with an increasing filtration; in particular,this implies that the product of Vassiliev invariants of degrees ≤ p and ≤ qis a Vassiliev invariant of degree no greater than p+ q.

Proof. This assertion is a direct corollary of Theorems 4.4.2 and A.2.19(see Appendix).

We have thus proved Theorem 3.6.1 announced earlier.

The algebra of Vassiliev invariants V is a subalgebra of I; in the termi-nology of Sec. A.2.13 it is nothing but the reduced part of I by Goussarov’sfiltration.

4.4.5. Exercise. Prove that Goussarov’s filtration in the space of knots isinfinite and, moreover, that the dimension of the quotient space X/ ∩∞n=0

Xn is infinite. Correspondingly, the filtration of the algebra of Vassilievinvariants by degree is infinite and dimI/V =∞.

4.5. Hopf algebra of chord diagrams

4.5.1. The vector space of chord diagrams. A dual way to define theweight systems is to introduce the 1- and 4-term relations directly in thevector space generated by chord diagrams.

4.5.2. Definition. The space An of chord diagrams of order n is the vec-tor space generated by the set CDn (all diagrams of order n) modulo thesubspace spanned by all 4-term linear combinations

− + − .

The space A′n of unframed chord diagrams of order n is the quotient of Anby the subspace spanned by all diagrams with an isolated chord.

In these terms, the space of weight systems Wn (resp. the space ofunframed weight systems W ′n) is dual to the space of chord diagrams An(resp. to the space of unframed chord diagrams A′n):

Wn = Hom(An,F),

W ′n = Hom(A′n,F).

4.5. Hopf algebra of chord diagrams 91

Below, we list the dimensions and some bases of the spaces An for n = 1,2 and 3:

A1 = 〈〉, dimA1 = 1.

A2 = 〈 , 〉, dimA2 = 2, because the only 4-term relation that

can be set up out of chord diagrams of order 2 is trivial.

A3 = 〈 , , 〉, dimA3 = 3, because #(CD3) = 5, and

there are 2 independent 4-term relations:

= and − 2 + = 0.

Taking into account the 1-term relations, we get the following result forthe spaces of unframed chord diagrams of small orders:

A′1 = 0, dimA′1 = 0.

A′2 = 〈〉, dimA′2 = 1.

A′3 = 〈〉, dimA′3 = 1.

The result of similar calculations for order 4 diagrams is presented inTable1. In this case dimA4 = 6; in the table, the set d4

3, d46, d

47, d

415, d

417, d

418

is used as a basis. The table is obtained by running Bar-Natan’s computerprogram available at [BN5]. The numerical notation for chord diagramslike [12314324] is easy to understand: one writes the numbers on the circlein the positive direction and connects equal numbers by chords. Among allpossible codes the lexicographically minimal one is used.

4.5.3. Multiplication of chord diagrams. Now we are ready to definethe structure of an algebra in the vector spaceA =

⊕k≥0

Ak of chord diagrams.

Definition. The product of two chord diagrams D1 and D2 is defined bycutting and gluing the two circles as shown here:

= =

(mind the orientations of the circles, which are counterclockwise by default).


CD Code and expansion CD Code and expansion

d41 = [12341234]

= d43 +2d4

6− d47− 2d4

15 + d417

d42 = [12314324]

= d43 − d4

6 + d47

d43 = [12314234]

= d43

d44 = [12134243]

= d46 − d4

7 + d415

d45 = [12134234]

= 2d46 − d4

7

d46 = [12132434]

= d46

d47 = [12123434]

= d47

d48 = [11234432]

= d418

d49 = [11234342]

= d417

d410 = [11234423]

= d417

d411 = [11234324]

= d415

d412 = [11234243]

= d415

d413 = [11234234]

= 2d415 − d4

17

d414 = [11232443]

= d417

d415 = [11232434]

= d415

d416 = [11223443]

= d418

d417 = [11223434]

= d417

d418 = [11223344]

= d418]

Table 1. Chord diagrams of order 4

This mapping is then extended by linearity to

µ : Am ⊗An → Am+n.

Remark. In the example above we could with an equal success cut thecircles in other points and get a different result:

Lemma. The result of multiplication is well-defined modulo 4T relations.

Proof. 1. In the case of the product of two diagrams, it is enough to provethat if one of the two diagrams, say D2, is turned inside the product diagramby one “click” with respect to D1, then the result is the same modulo 4Trelations.

4.5. Hopf algebra of chord diagrams 93

Note that this turn is equivalent to the following transformation. Picka chord in D2 with endpoints a and b such that a is adjacent to D1. Then,fixing the endpoint b, move a all through the diagram D1. In the process ofthis motion we obtain 2n+ 1 diagrams P0, P1, ..., P2n, where n is the orderof D1, and we must prove that P0 ≡ P2n mod 4T . Now, if you think a littlebit, you will see that the difference P0 − P2n is in fact equal to the sum ofall n four-term relations which are obtained by fixing the endpoint b andall chords of D1, one by one. For example, if we consider the two productsshown above and use the following notation:

b

a

b

a

b

a

b

a

ba

baa

b

P0 P1 P2 P3 P4 P5 P6

then we must take the sum of the three linear combinations

P0 − P1 + P2 − P3,

P1 − P2 + P4 − P5,

P3 − P4 + P5 − P6,

and the result is exactly P0 − P6.

2. Now we must prove that the product of any two linear combinationsof chord diagrams is well-defined modulo 4T relations. This fact directlyfollows from 1 and standard linear algebra.

4.5.4. Comultiplication of chord diagrams. Here, we will introducethe coproduct in A and thus turn this algebra into a bialgebra.

Definition. The coproduct in the algebra A

δ : An →⊕

k+l=n

Ak ⊗Al

is defined as follows. For a diagram D ∈ An we put

δ(D) :=∑

J⊆[D]

DJ ⊗DJ ,

the summation taken over all subsets J of the set of chords of D. Here DJ

is the diagram consisting of the chords that belong to J and J = [D] \ J isthe complementary subset of chords. To the entire space A the operator δis extended by linearity.

If D is a diagram of order n, then the total number of summands in theright hand side of the definition is 2n.


Example.

δ( )

= ⊗ + ⊗ + ⊗ + ⊗

+ ⊗ + ⊗ + ⊗ + ⊗

= ⊗ + 2 ⊗ + ⊗

+ 2 ⊗ + ⊗ + ⊗

Lemma. The coproduct δ is well-defined modulo 4T relations.

Proof. Unlike the similar property of multiplication, this fact is fairly evi-dent.

The unit and the counit in A are defined as follows:

ι : F→ A , ι(x) = x ,

ε : A → F , ε(x + ...) = x.

Exercise. Check the axioms of a bialgebra.

4.5.5. Deframing of chord diagrams. The space of unframed chord di-agrams A′ was defined as the quotient of the space A over the subspacespanned by all diagrams with an isolated chord. In terms of multiplica-tion just introduced in A, this subspace can be described as the ideal of Agenerated by Θ, the chord diagram with one chord, so that we can write:

A′ = A/(Θ).

It turns out that there is a simple explicit formula for a linear operatorp : A → A whose kernel is the ideal (Θ). Namely, define pn : An → An by

pn(D) :=∑

J⊆[D]

(−Θ)n−|J | ·DJ ,

where, as earlier, [D] stands for the set of chord in the diagram D and DJ

means the subdiagram of D with only the chords from J left. The sum ofpn over all n is the operator p : A → A.

4.5.6. Exercise. Check that

(1) p is a homomorphism of algebras,

4.6. Hopf algebra of weight systems 95

(2) p(Θ) = 0 and hence p takes the entire ideal (Θ) into 0.

(3) p is a projector, i.e. p2 = p.

(4) the kernel of p is exactly (Θ).

We see, therefore, that the quotient map p : A/(Θ) → A is the iso-morphism of A′ onto its image and we have a direct decomposition A =p(A′) ⊕ (Θ). Note that the first summand here is different from the sub-space spanned merely by all diagrams without isolated chords!

For example, p(A3) is spanned by the two vectors

p( ) = − 2 + ,

p( ) = − 3 + 2 .

and its dimension is 1, while the subspace generated by the elements

and themselves is 2-dimensional and has a nonzero intersection with

the ideal (Θ).

4.6. Hopf algebra of weight systems

In Section 4.5.1, we defined weight systems as elements of the vector spaceW, dual to the space A. Now that A is equipped with the structure of a Hopfalgebra, the general construction of Section A.2.23 supplies the spaceW withthe same structure, too. In particular, weight systems can be multipliedtogether. For example, if w1 is a weight system which takes value a on the

chord diagram , and zero value on all other chord diagrams, and w1

takes value b on and vanishes elsewhere, then

(w1 · w1)( ) = (w1 ⊗ w2)(δ( )) = 2w1( ) · w2( ) = 2ab .

A weight system that belongs to a homogeneous component Wn of thespace W is called homogeneous of degree n. A simple example of a homo-geneous weight system of degree n is provided by the function on the setof chord diagrams equal to 1 on any diagram of degree n and vanishing onchord diagrams of degree different from n. The four-term relations for thisfunction evidenttly holds. Let us denote this weight system by 1n.

4.6.1. Lemma. 1n · 1m =(m+nn

)1n+m.


This directly follows from the definition of multiplication of weight sys-tems.

4.6.2. Corollary. (i) 1n1n! = 1n;

(ii) e11 :=∞∑

n=0

1

n!1n1 =

∞∑

n=0

1n =: 1, where 1 is a weight system that is

equal to 1 on every chord diagram.

Strictly speaking, 1 is not an element of W =⊕nWn because it has

infinitely many nonzero terms. Formally, it is an element of the gradedcompletion W whose elements are represented by formal infinite series ofelements of Wn over all n. The space W is a (completed) Hopf algebra.Please note that 1 is not the unit of W. Its unit, as well as the unit of Witself, is represented by the element 10 — the weight system which takesvalue 1 on the chord diagram without chords and vanishes elsewhere.

Let w ∈ W be an element with homogeneous components wi ∈ Wi suchthat w0 = 0. Then the exponent of w can be defined as the Taylor series

ew =∞∑

k=1

wk

k!.

This formula makes sense because for the evaluation of each homogeneouscomponent of this sum only a finite number of operations is required. Forexample, if θ ∈ F is a number (element of the ground field), then we may

consider the weight system eθ11 . It takes value θn an any chord diagram

with n chords. One can easily check that the weight systems eθ11 and e−θ11

are inverse to each other:

eθ11 · e−θ11 = 10 .

4.6.3. Deframing of weight systems. Since A′ = A/(Θ) is a quotient ofA, the dual spaces are embedded one into another W ′ ⊂ W. The elementsof W ′ take zero values on all chord diagrams with an isolated chord. InSection 4.1 they were called unframed weight systems.

The deframing procedure for chord diagrams (Section 4.5.5) leads to adeframing procedure for weight systems. By duality, the projector p : A →A gives rise to a projector p∗ :W →W whose value on an element w ∈ Wn

is defined by

w′ = p∗(w)(D) := w(p(D)) =∑

J⊆[D]

w((−Θ)n−|J | ·DJ

).

Obviously, w′(D) = 0 for any w and any chord diagram D with an isolatedchord. So the operator p∗ : w 7→ w′ is a projection of the space W onto itssubspace W ′ consisting of unframed weight systems.

4.7. Primitive elements in A 97

The deframing operator looks especially nice for multiplicative weightsystems. We call a weight system w multiplicative if for any two chorddiagrams D1 and D2 we have w(D1 ·D2) = w(D1) ·w(D2). This is the sameas to say that w is a group-like element of the Hopf algebra W.

4.6.4. Exercise. Prove that for any number θ ∈ F the exponent eθ11 ∈Wis a multiplicative weight system.

Lemma. Let θ = w(Θ) for a multiplicative weight system w. Then its

deframing is w′ = e−θ11 · w.

We leave the proof of this lemma to the reader as an exercise. Thelemma, together with the previous exercise, implies that the deframing of amultiplicative weight system is again multiplicative.

4.7. Primitive elements in ASince the algebra of chord diagrams A satisfies the premises of theoremA.2.24, it also satisfies its implication. This means that any element of A isrepresented as a polynomial in basic primitive elements. Let us denote then-th homogeneous component of the primitive subspace by Pn = An∩P (A)and find an explicit description of Pn for small n.

dim = 1. P1 = A1 is one-dimensional and generated by .

dim = 2. Since

δ( ) = ⊗ + 2 ⊗ + ⊗ ,

δ( ) = ⊗ + 2 ⊗ + ⊗ ,

the element − is primitive. It constitutes a basis of P2.

dim = 3. The coproducts of the 3 basic elements of A3 are

δ( ) = ⊗ + 2 ⊗ + ⊗ + . . . ,

δ( ) = ⊗ + ⊗ + 2 ⊗ + . . . ,

δ( ) = ⊗ + 3 ⊗ + . . .

(Here the dots designate the terms, symmetric to the terms that are writtenout.) Looking at these expressions, it is easy to check that the element

− 2 + is the only primitive homogeneous element of A3.


The exact dimensions of Pn are now known up to n = 12 (the last threevalues, corresponding to n = 10, 11, 12 were found by J. Kneissler [Kn0]):

n 1 2 3 4 5 6 7 8 9 10 11 12

dimPn 1 1 1 2 3 5 8 12 18 27 39 55

We will discuss the sizes of the spaces Pn, An and Vn in more detaillater (see section 14.4).

Exercise. Find a basis in the primitive space P4.

If the dimensions of Pn were known for all n, then the dimensions of Anwould also be known.

Example. Let us find the dimensions of An, n ≤ 5, assuming that weknow the values of dimPn for n = 1, 2, 3, 4, 5, which are equal to 1, 1, 1, 2, 3.Let pi be the basic element of Pi, i = 1, 2, 3 and denote the bases of P4

and P5 as p41, p42 and p51, p52, p53, respectively. Nontrivial monomials upto degree 5 that can be made out of these basic elements are:

Degree 2 monomials (1): p21.

Degree 3 monomials (2): p31, p1p2.

Degree 4 monomials (4): p41, p

21p2, p1p3, p

22.

Degree 5 monomials (7): p51, p

31p2, p

21p3, p1p

22, p1p41, p1p42, p2p3.

A basis of each An can be made up of the primitive elements andtheir products of the corresponding degree. For n = 0, 1, 2, 3, 4, 5 we get:dimA0 = 1, dimA1 = 1, dimA2 = 1 + 1 = 2, dimA3 = 1 + 2 = 3,dimA4 = 2 + 4 = 6, dimA5 = 3 + 7 = 10.

The partial sums of this sequence give the dimensions of the spaces offramed Vassiliev invariants: dimFV0 = 1, dimFV1 = 2, dimFV2 = 4,dimFV3 = 7, dimFV4 = 13, dimFV5 = 23.

Note that primitive elements of A are represented by rather complicatedlinear combinations of chord diagrams. A more concise and clear represen-tation can be obtained through connected closed diagrams, to be introducedin the next chapter.

4.8. Linear chord diagrams

If, instead of conventional closed knots one considers long knots (see 1.8.3),then the same considerations as we used in this chapter lead naturally tothe space of linear chord diagrams, i.e. diagrams on an oriented line:

4.8. Linear chord diagrams 99

subject to the 4-term relations:

− + − = 0.

In words, the rule of forming a 4-term relation reads as follows: themoving endpoint of the additional chord is placed twice in front of theimmobile chord’s endpoints, and these terms are taken with a plus sign,then it is twice placed behind that chord’s endpoints, and these terms comewith a minus.

Let us temporarily denote the space of linear chord diagrams of order nby Aln; we have, like before: Aln = 〈CDl

n〉/〈4T ln〉.If the line is closed into a circle, linear 4-term relations evidently become

circular 4-term relations, and we arrive at a linear map Aln → An. This mapis evidently onto, because one can find a preimage of every circular CD bycutting the circle at an arbitrary point. However, it may seem that this maphas a kernel, because the preimage, in general, depends on the place wherethe circle is cut. For example, the linear diagram shown above has the sameimage in CD3 as the one drawn below:

Remarkably, modulo four-term relations, all preimages of any circularchord diagram are equal (in particular, the two sample diagrams in theabove pictures are equal in Al3). This fact is proved by exactly the sameprocedure as the one used in the proof that multiplication of chord diagramsis well-defined (Lemma 4.5.3): one takes all ‘immobile’ chords one by oneand for each of them writes out a 4-term relation for an additional movingchord. In the sum of all these combinations each term, except for just two,appears twice with different signs, and we are left with two diagrams: withthe leftmost and rightmost positions of the moving endpoint.

This implies:

Proposition. The natural map of closing a linear chord diagram into acircle gives rise to a vector space isomorphism Al → A.

In fact, chord diagrams with 4-term relations can be considered on an ar-bitrary one-dimensional orientable manifold — see Section 8.6.2 and Chapter11.


Exercises

(1) A short chord is a chord whose endpoints are adjacent, i.e. one ofthe arcs that it subtends contains no endpoints of other chords. Inparticular, short chords are isolated. Prove that the linear span of alldiagrams with a short chord and four-term combinations contains alldiagrams with an isolated chord. This means that the restricted one-term relations (only for diagrams with a short chord) imply generalone-term relations provided that four-term relations hold.

(2) Find the number of different chord diagrams of order n with n isolatedchords. Prove that all of them are equal to each other modulo the four-term relations.

(3) Using Table1, find the space of unframed weight systems W ′4.Answer. The basic weight systems are:

1 0 -1 -1 0 -1 -2

0 1 1 2 0 1 3

0 0 0 0 1 1 1

The table shows that the three diagrams , and form

a basis in the space A′4.(4) ∗ Is it true that any chord diagram of order 13 is equivalent to its mirror

image modulo 4-term relations?

(5) Prove that the deframing operator ′ (Sec.4.6.3) is a homomorphism ofalgebras: (w1 · w2)

′ = w′1 · w′2.(6) Give a proof of the Lemma from Sec.4.6.3.

(7) Prove that for any primitive element p of degree > 1, w(p) = w′(p)where w′ is the deframing of an arbitrary weight system w.

(8) Given the generating function for the sequence of dimensions of primitivespaces in a polynomial algebra

p(t) =∑

n≥0

(dimPn)tn,

find the generating function for the sequence of dimensions of the entirealgebra

a(t) =∑

n≥0

(dimAn)tn.

Exercises 101

(9) Describe group-like elements in the completed Hopf algebra of chorddiagrams A.

(10) Prove that the symbol of a primitive Vassiliev invariant is a primitiveweight system.

(11) Let Θ be a chord diagram with a single chord. By a direct computation,check that exp(αΘ) :=

∑∞n=0

αnΘn

n! ∈ A is a group-like element in thecompleted Hopf algebra of chord diagrams.

Part 2

Diagrammatic

Algebras

Chapter 5

Jacobi diagrams

In the previous chapter we saw that the study of Vassiliev knot invariantsis largely reduced to the study of the algebra of chord diagrams. Herewe introduce other related types of diagrams, called closed Jacobi diagramsand open Jacobi diagrams that provide better understanding of the primitivespace P(A) and bridge the way to the Lie algebras (chapter 7), the wheelingtheorem (10.6) and the Drinfeld associator (chapter 12).

The word Jacobi is justified by a close resemblance of the basic relatonsimposed on Jacobi diagrams (STU and IHX) with the Jacobi identity forLie algebras.

5.1. Closed Jacobi diagrams

A chord diagram can be considered as a graph consisting of n disconnectedsegments (graphs with two vertices and one edge), attached to a circle.Closed Jacobi diagrams generalize this construction in that they are alsoallowed to have trivalent vertices in the graph attached to the circle. Otherterms used for the same object in the literature include Chinese charac-ter diagrams [BN1], circle diagrams [Kn0], round diagrams [Wil1] andFeynman diagrams [KSA]. Closed diagrams are considered modulo certainrelations; in the quotient space, each closed diagram equals a linear combi-nation of chord diagrams and can be viewed as a short-hand notation forthat combination.

Modulo the relations, the space of closed diagrams becomes a bialgebrawhich is isomorphic to the algebra of chord diagrams A, and in the futurewe will not distinguish between the two.

105

106 5. Jacobi diagrams

5.1.1. Definition. A closed Jacobi diagram of degree n is a connectedthree-valent graph on 2n vertices with a distinguished oriented cycle, calledWilson loop, and a fixed rotation (cyclic order of edges) in all the verticesthat do not belong to the Wilson loop.

Example. Here is a closed diagram of degree 4:

Orientation of the Wilson loop, as well as the rotation of the inner vertices,is indicated by arrows. In the pictures below, we will always draw thediagram inside its Wilson loop, which is oriented counterclockwise unlessotherwise specified. Inner trivalent vertices will be also supposed to beoriented countercloskwise. (This convention is referred to as the blackboardorientation.) Note that the intersection of two edges in the center of thediagram is not actually a vertex.

5.1.2. Definition. The space of closed diagrams C is the direct sum of thespaces Cn for all n ≥ 0. Each Cn is the vector space spanned by all closeddiagrams of degree n modulo STU relations:

S

=

T

−U

.

The three diagrams S, T and U must be identical outside the shownfragment.

The choice of a specific STU relation is determined by the diagram onthe left and one of its internal trivalent vertices adjacent to the Wilson loop.The signs of the T- and U-terms are determined by the orientation of theWilson loop and the cyclic order of the three edges meeting at the internalvertex of the S-term. In the shown figure they are both conterclockwise.Should we reverse one of them, say the orientation of the Wilson loop, thesigns on the right become opposite. Indeed,

=STU= − = − .

This remark will be important in Section 5.5.3 when we discuss the problemdetecting knot orientation. Also one may think about this as a choice of the

5.1. Closed Jacobi diagrams 107

cyclic order ‘forward-sideways-backwards’ at a vertex lying on the Wilsonloop. Then the decision about signs in the STU relation is based on thecyclic orders at both vertices of the S-term. In this case the relation abovemay be understood as a consequence of the anisymmetry relation AS (see5.2.2) for the vertex on the Wilson loop, and the STU relation itself regardedas a particular case of the IHX relation (see 5.2.3).

5.1.3. Examples. There exist two different closed diagrams of order 1:

, , one of which vanishes due to the STU relation:

= − = 0 .

There are four non-zero closed diagrams of degree 2:

, , ,

whereas the STU relations ensure that dim C2 ≤ 2.

Indeed, by STU relations:

= − ;

= − = 2 .

Note, moreover, that there are four more closed diagrams of degree 2,which are 0 due to the AS relations (see Lemma 5.2.5 below). The first of

these relations gives a concise representation, , of the basic primitive

element of degree 2.

5.1.4. Exercise. Using the STU relations, rewrite the basic primitive ele-ment of order 3 in a concise way.

Answer.

− 2 + = .

Chord diagrams are a particular case of closed diagrams, namely thosewithout inner trivalent vertices. Using STU relations, one can rewrite anyclosed diagram as a linear combination of chord diagrams. (Examples weregiven just above.)


A vertex of a closed diagram that belongs to the Wilson loop is calledexternal; otherwise it is called internal. In the space Cn there is an increasingfiltration by subspaces Cmn generated by diagrams with at most m externalvertices:

C1n ⊂ C2

n ⊂ ... ⊂ C2nn .

Exercise. Prove that C1n = 0.

Hint. This easily follows from the STU relation.

5.2. IHX relation

5.2.1. Lemma. The STU relations imply the 4T relations for chord dia-grams.

Proof. Indeed, if we write the four-term relation under the form

− = −

and apply the STU relations to the left and to the right part of this equation,we will get the same closed diagrams.

5.2.2. Definition. An AS (=antisymmetry) relation is:

= − ,

that is, a diagram changes sign when the cyclic order of three edges at atrivalent vertex is reversed.

5.2.3. Definition. An IHX relation is:

= − .

(You may notice that the three fragments involved resemble the capitalletters I, H and X, respectively).

As always, the unfinished fragments of the pictures denote graphs thatare identical (and arbitrary) everywhere but in this explicitly shown frag-ment.

5.2.4. Exercise. Check that the three terms of the IHX relation have equalrights. For example, look at the H sideways, then it becomes an I; write anIHX relation starting from that I and make sure it is the same as the initialone. Also, single out a portion of the X which looks like an H, write out anIHX relation with that H and make sure it is again the same.

5.2. IHX relation 109

5.2.5. Lemma. The STU relations imply the AS relations for the internalvertices of a closed Jacobi diagram.

Proof. Induction on the distance (in edges) of the vertex in question fromthe Wilson loop.

Induction base. If the vertex is adjacent to an external vertex, then theassertion follows by one application of the STU relation:

= −

= − .

Induction step. Take two closed diagrams f1 and f2 that differ only bythe rotation of one internal vertex v. Apply STU relations to both diagramsin the same way so that v gets closer to the Wilson loop.

The lemma is proved.

5.2.6. Lemma. The STU relations imply the IHX relations for the internalvertices of a closed diagram.

Proof. The argument is similar to the one used in the previous proof. Wetake an IHX relation somewhere inside a closed diagram and, applying thesame sequence of STU moves to each of the three diagrams, move the IHXfragment closer to the Wilson loop. The proof of the induction base is shownin these pictures:

= − = − − + ,

= − = − − + ,

= − = − − + .

Therefore,

= + ,

which is the IHX relation that we wanted to prove.


5.2.7. Other forms of the IHX relation. The IHX relation can be re-drawn under several different forms, for example:

• (rotationally symmetric form)

+ + = 0 .

• (Jacobi form)

= + .

• (Kirchhoff form)

= + .

5.2.8. Exercise. By moving your head and pulling on the strings of thediagrams, make sure that all these forms are equivalent.

The Jacobi form of the IHX relation can be interpreted as follows. Sup-pose that to the upper 3 endpoints of each diagram we assign 3 elements ofa Lie algebra, x, y and z, while every trivalent vertex, traversed downwards,takes the pair of ‘incoming’ elements into their commutator:

x y

[x, y]

.

Then the IHX relation means that

[[x, y], z]− [x, [y, z]]− [[x, z], y] = 0,

which is the classical Jacobi identity. This observation, properly developed,leads to the construction of Lie algebra weight systems — see Chapter 7.

The Kirchhoff presentation is reminiscent of the Kirchhoff’s law in elec-

tricity. Let us view the portion of the given graph as a piece of

electrical circuit, and the variable vertex as an ‘electron’ e with a ‘tail’ whoseendpoint is fixed. Suppose that the electron moves towards a node of thecircuit:

e

“tail”

5.2. IHX relation 111

Then the IHX relation expresses the well-known Kirchhoff rule: the sum ofcurrents entering a node is equal to the sum of currents going out of it. Thiselectrotechnical analogy is very useful, e. g. in the proof of the generalizedKirchhoff law below.

The IHX relation can be generalized as follows:

5.2.9. Lemma. (Generalized Kirchhoff rule). The following identity holds:

12···k

=

k∑

i=1

1···i···k

,

where the grey box is an arbitrary subgraph which has only 3-valent vertices.

Proof. Fix a horizontal line in the plane and consider an immersion of thegiven graph into the plane with smooth edges, generic with respect to theprojection onto this line. More exactly, we assume that (1) the projectionsof all vertices onto the horizontal line are distinct, (2) when restricted toan arbitrary edge, the projection has only Morse critical points, and (3) theimages of all critical points are distinct and different from the images ofvertices.

Bifurcation points are the images of vertices and critical points of theprojection. The proof proceeds by induction on the number of bifurcationpoints. If there is only one bifurcation points, it must be the image ofa vertex, in which case the equality to be proved coincides with the IHXrelation. Suppose that the lemma is established for s bifurcation points.The (s+ 1)-th bifurcation point falls under one of the six categories:

1) 2) 3) 4) 5) 6) .

In the first two cases the lemma follows from the IHX relation, in cases 3 and4 — from the AS relation. Cases 5 and 6 by a deformation of the immersionare reduced to a combination of the previous cases (also, they can be doneby one application of the IHX relation in the symmetric form).

This completes the proof of the lemma.


Example.

= +

= + + +

= + + +

The two last summands canceleach other because of the antisymmetry re-lation.

5.3. Isomorphism A ≃ CLet CDn be the set of chord diagrams of order n and CDn the set of closeddiagrams of the same order. We have a natural inclusion λ : CDn → CDn.

5.3.1. Theorem. The inclusion λ gives rise to an isomorphism of vectorspaces λ : An → Cn.

Proof. Indeed, we must check the following things:

(A) That λ leads to a well-defined linear mapping from An to Cn.(B) That this mapping is a linear isomorphism.

Well, let’s see.

(A) Indeed, An = 〈CDn〉/〈4T〉, Cn = 〈CDn〉/〈STU〉. (Remind thatangular brackets denote linear span.) Lemma 5.2.1 implies that λ(〈4T〉) ⊂〈STU〉), therefore the map of the quotient spaces is well-defined.

(B) We will construct a linear mapping ρ : Cn → An and prove that itis inverse to λ.

As we noticed before, any closed diagram by iterative use of STU rela-tions can be transformed into a combination of chord diagrams. This givesrise to a mapping ρ : CDn → 〈CDn〉 which is, however, multivalued, becausethe result may depend on the specific sequence of resolutions used. Here is

5.3. Isomorphism A ≃ C 113

an example where the place where the STU relation is applied is marked:

* 7→ − = 2 7→ 2(

− 2 +),

*

7→ − = 2*

− 2 *

7→ 2(

− − +).

However, the combination ρ(C) is well-defined as an element of An, i. e.modulo the 4T relations. The proof of this fact proceeds by induction onthe number k of inner points in the diagram C.

If k = 1, then the diagram C consists of one tripod and several chordsand may look something like this:

There are 3 ways to resolve the inner triple point, and the fact that theresults are the same in An is exactly the definition of the 4T relation.

Suppose that ρ is well-defined on closed diagrams with< k inner vertices.

Pick a diagram C ∈ CD2n−kn . The process of resolving the triple points starts

with some 2 out of 2n−k outer vertices. Let us prove, modulo the inductivehypothesis, that if we change the order of these two points, then the finalresult will be the same.

There are 3 cases to consider: the two chosen points on the Wilson loopare (1) adjacent to a common inner vertex, (2) adjacent to neighboring innervertices, (3) adjacent to non-neighboring inner vertices. The proof for cases(1) and (2) is shown in the pictures that follow.

(1)

∗7−→ − ,

∗7−→ − ,

The position of an isolated chord does not matter, because, as we know, themultiplication in A is well-defined (as well as in C, see the next section).

(2)


∗7−→

∗−

∗

7−→ − − + ,

∗7−→

∗−∗

7−→ − − + .

After the first resolution, we can choose the sequence of further resolutionsarbitrarily, by the inductive hypothesis.

Exercise. Write up a similar proof in case (3).

We thus have a well-defined linear mapping ρ : Cn → An. The fact thatit is two-sided inverse to λ is fairly evident. The theorem is proved.

5.4. Product and coproduct in CNow we will define a bialgebra structure in the space C.5.4.1. Definition. The product of two closed diagrams is defined in thesame way as for chord diagrams: the two Wilson loops are cut at arbitraryplaces and then glued up into one loop, in agreement with the orientations:

· = .

5.4.2. Proposition. This multiplication is well-defined, i. e. does not de-pend on the place of cuts.

Proof. Due to the 4T relations in A and the STU relations in C, we seestraight away that this product in C agrees with the product in A along theisomorphism A ≃ C constructed in Theorem 5.3.1 above. Now the fact thatthe multiplication is well-defined in A implies the fact that it is well-definedin C.

To define the coproduct in the space C, we will use connected componentsof a closed diagram.

By definition, the underlying graph of a closed diagram is always con-nected. We will, however, introduce a different notion of connectivity anddistinguish between connect and disconnected closed diagrams.

5.5. Primitive subspace of C 115

5.4.3. Definition. A closed diagram is said to be connected if its underlyinggraph remains connected after stripping the Wilson loop. The connectedcomponents of a closed diagram are defined as connected components of thegraph obtained when the Wilson loop is removed.

In the sense of this definition, any chord diagram of order n consists ofn connected components — the maximal possible number.

Now, the construction of coproduct proceeds in the same way as forchord diagrams.

5.4.4. Definition. Let D be a closed diagram and [D] the set of its con-nected components. For any subset J ⊆ [D] denote by DJ the closed dia-gram with only those components that belong to J and by DJ the ‘comple-

mentary’ diagram (J := [D] \ J). We set by definition

δ(D) :=∑

J⊆[D]

DJ ⊗DJ .

Example:

δ( )

= 1⊗ + ⊗ + ⊗ + ⊗1.

We know that the algebra C, as a vector space, is spanned by chorddiagrams. For chord diagrams, algebraic operations defined in A and C,tautologically coincide. It follows that the coproduct in C is consistent withits product and that the isomorphisms λ, ρ are in fact isomorphisms ofbialgebras.

5.5. Primitive subspace of CBy definition, connected closed diagrams are primitive with respect to thecoproduct δ. Taking into account the sad situation with chord diagrams, itmay sound surprising that the converse is also true:

5.5.1. Theorem. [BN1] The primitive space P of the bialgebra C coincideswith the linear span of connected closed diagrams.

Proof. If the primitive space P were bigger than the span of connectedclosed diagrams, then, according to Theorem A.2.24, it would contain anelement that cannot be represented as a polynomial of connected closeddiagrams. Therefore, to prove the theorem it is enough to show that everyclosed diagram is a polynomial of connected diagrams. This can be doneby induction on the number of external vertices (legs) of a closed diagramC. Suppose that the diagram C consicts of several connected components(see 5.4.3). The STU relation tells us that we can freely interchange the


legs of C modulo closed diagrams with fewer legs. Using such permutationswe can separate the connected components of C. This means that moduloclosed diagrams with fewer legs C is equal to the product of its connectedcomponents.

5.5.2. Filtration of Pn. The primitive space Pn cannot be graded bythe number of external vertices (legs) k, because the STU relation is nothomogeneous with respect to k. However, it can be filtered :

0 = P1n ⊆ P2

n ⊆ P3n ⊆ · · · ⊆ Pn+1

n = Pn .

where Pkn be the subspace of Pn generated by connected closed diagramswith at most k external vertices.

The connectedness of a closed diagram with 2n vertices implies that thenumber of its external vertices cannot be bigger than n + 1. That is whythe filtration ends at the term Pn+1

n .

The following facts about the filtration are known.

• [CV] The filtration ends even earlier. Namely, Pnn = Pn for evenn, and Pn−1

n = Pn for odd n. Moreover, for even n the quotientspace Pnn/Pn−1

n has dimension one and is generated by the wheelwn with n spokes:

wn =

n spokes

This fact has relation to the Melvin-Morton conjecture (see Section14.1 and Exercise 11).

• [Da1] The quotient space Pn−1n /Pn−2

n has dimension [n/6] + 1 forodd n, and 0 for even n.

• [Da2] For even n

dim(Pn−2n /Pn−3

n ) =

[(n− 2)2 + 12(n− 2)

48

]+ 1 .

• For small degrees the dimensions of the quotient spaces Pkn/Pk−1n

were calculated by J. Kneissler [Kn0] (empty entries in the tableare zeroes):

5.5. Primitive subspace of C 117

ccnk 1 2 3 4 5 6 7 8 9 10 11 12 dimPn

1 1 1

2 1 1

3 1 1

4 1 1 2

5 2 1 3

6 2 2 1 5

7 3 3 2 8

8 4 4 3 1 12

9 5 6 5 2 18

10 6 8 8 4 1 27

11 8 10 11 8 2 39

12 9 13 15 12 5 1 55

5.5.3. Detecting the knot orientation. One may notice that in the tableabove all entries with odd k are zeroes. This means that any connectedclosed diagram with an odd number of legs is equal to a suitable linearcombination of diagrams with fewer legs. This observation is tightly relatedto the problem of distinguishing knot orientation by Vassiliev invariants.The existence of the universal Vassiliev invariant given by the Kontsevichintegral reduces the problem of detecting the knot orientation to a purecombinatorial problem. Namely, Vassiliev invariants do not distinguish theorientation of knots if and only if every chord diagram equals itself withthe reversed orientation of the Wilson loop modulo 4T relations (see theCorrolary in 9.3). If we denote the operation of reversing the orientation byS, then its action on a chord diagram D is equivalent to the axial reflectionof the diagram as a planar picture, and the question is whether D = S(D) inA. The followng Theorem translates this fact into the language of primitivesubspaces.

Theorem. Vassiliev invariants do not distinguish the orientation of knotsif and only if Pkn = Pk−1

n for odd k and arbitrary n.

To prove the Theorem we need to reformulate the question whetherD = S(D) in terms of closed diagrams. Reversing the orientation of theWilson loop on closed diagrams should be done with some caution, becauseof the discussion in 5.1.2 (see p. 106). What we do is we carry the operationS from chord diagrams to closed diagrams along the isomorphism λ : A → Cand then we have the following assertion.


Lemma. Let P = P ′ be a closed diagram with k external vertices.

Then

S(P ) = (−1)k P ′ .

Proof. Indeed, represent P as a linear combination of chord diagrams us-ing STU relations, then reverse the orientation of the Wilson loop of allchord diagrams of the combination. After that, convert the obtained linearcombination back to a closed diagram. Now each application of the STUrelation multiplies the result by −1 because of the reversed Wilson loop(see page 106). All together we have to perform the STU relation 2n − ktimes, where n is the degree of P . Therefore the result gets multiplied by(−1)2n−k = (−1)k.

In the particular case k = 1 the Lemma asserts that P1 = 0, i.e. anydiagram with only one external vertex is 0 modulo the STU relations.

The operation S : C → C is in fact an algebra automorphism, S(C1·C2) =S(C1) ·S(C2). Therefore, to check the equality S = idC it is enough to checkit on the primitive subspace, i. e. determine whether P = S(P ) for everyconnected closed diagram P .

Corollary of the Lemma. Let P ∈ Pk =∞⊕n=1Pkn be a connected closed

diagram with k legs. Then S(P ) ≡ (−1)kP mod Pk−1.

Proof of the Corollary. Rotating the Wilson loop in 3-space by 180

about the vertical axis, we get:

S(P ) = (−1)k P ′ = (−1)k P ′

.

The STU relations allow us to permute the legs modulo diagrams with fewernumber of legs. Doing the corresponding permutation on the last diagramwe can straighten out all legs and get (−1)kP .

Proof of the Theorem. Suppose that the Vassiliev invariants do notdistinguish the orientation. Then S(P ) = P for every connected closeddiagram P . In particular, for a diagram P with an odd number of legs k wehave P ≡ −P mod Pk−1. Hence 2P ≡ 0 mod Pk−1, which means that Pis equal to a linear combination of diagrams with fewer legs, and thereforedim(Pkn/Pk−1

n ) = 0.

5.6. Open Jacobi diagrams 119

Conversely, suppose that Vassiliev invariants do distinguish the orienta-tion. Then there is a connected closed diagram P such that S(P ) 6= P . Pickup such P with the smallest possible number of legs k. Show that k cannotbe even. Consider X = P −S(P ) 6= 0. Since S is an involution S(X) = −X.But, in the case of even k, the non-zero element X has fewer legs than k,and S(X) = −X 6= X. So k cannot be the smallest in this case. Thereforesuch minimal k is odd, and dim(Pkn/Pk−1

n ) 6= 0.

5.6. Open Jacobi diagrams

The subject of this section is the combinatorial bialgebra B which is iso-morphic to the previously defined algebras A and C as linear space and ascoalgebra, but has a different natural multiplication. This leads to a re-markable fact that in our algebra A ≃ C ≃ B there are two multiplicationsboth compatible with one and the same comultiplication.

5.6.1. Definition. An open Jacobi diagram is a graph with 1- and 3-valentvertices, cyclic order of (half)edges at every 3-valent vertex and with at leastone 1-valent vertex in every connected component.

An open diagram is not required to be connected. It may have loopsand multiple edges. We will see later that, modulo the natural relationsany diagram with a loop vanishes. However, it is important to include thediagrams with loops in the definition, because the loops may appear duringnatural operations on open diagrams, and it is exactly because of this factthat we introduce the cyclic order on halfedges, not whole edges.

The total number of vertices of an open diagram is even. Half of thisnumber is called the degree (or order) of an open diagram. We will denotethe set of all open diagrams of degree n by ODn.

In the literature, open diagrams are also referred to as “1-3-valent dia-grams”, “Jacobi diagrams”, “web diagrams” and “Chinese characters”.

Examples. Below is the (hopefully complete) list of open diagrams ofdegree 1 and 2.

OD1 = ,

OD2 =

, , , , , ,

,

Now you will see that most of the elements listed above are not impor-tant: they are killed by the following definition.


5.6.2. Definition. The space of open diagrams of degree n is the quotientspace

Bn = 〈ODn〉/〈AS, IHX〉,where 〈ODn〉 is the vector space formally generated by all open diagrams ofdegree n and 〈AS, IHX〉 stands for the subspace spanned by all AS and IHXrelations (see 5.2.2, 5.2.3). By definition, B0 is one-dimensional, spanned bythe empty diagram.

Relations AS and IHX, unlike STU, preserve the separation of verticesinto 1- and 3-valent. Therefore, the space B is much more graded than A.Apart from the main grading by half the number of vertices, indicated bythe subscript in B, it also has a grading by the number of univalent vertices

B =⊕

n

⊕

k

Bkn,

where Bkn the subspace spanned by all diagrams with 2n vertices, amongwhich k univalent vertices (legs).

For disconnected diagrams the second grading can, in its turn, be refinedto a multigrading by the numbers of legs in each connected component ofthe diagram:

B =⊕

n

⊕

k1,...,km

Bk1,...,kmn .

Exercise. Write out all relations for n = 1 and 2 and show directly thatdimB1 = 1, dimB2 = 2.

Both the product and the coproduct in the linear space B are definedin a quite straightforward way. We first define the product and coproductof diagrams themselves, then extend it by linearity to the free vector spacespanned by the diagrams, then note that it is compatible with the AS andIHX relations and thus descends to the quotient space B.

5.6.3. Definition. The product of two open diagrams is their disjointunion.

Example: · = .

5.6.4. Definition. Let D be an open diagram and [D] the set of its con-nected components. For a subset J ⊆ [D] denote by DJ the union of thecomponents that belong to J and by DJ the union of the components thatdo not belong to J . We set

δ(D) :=∑

J⊆[D]

DJ ⊗DJ .

5.7. Linear isomorphism B ≃ C 121

Example:

δ( )

= 1⊗ + ⊗ + ⊗ + ⊗ 1,

Because the relations in B do not intermingle different connected compo-nents of a diagram, the product of an AS or IHX combination of diagrams byan arbitrary open diagram belongs to the linear span of the relations of thesame types. Also, the coproduct of any AS or IHX relation is also 0 modulothese relations. Therefore, we have well-defined algebraic operations in thespace B, and they are evidently compatible with each other. The space Bthus becomes a Hopf algebra.

5.6.5. Some identities. The relations AS and IHX imply many interest-ing identites between the elements of the space B. Some of them are provedin the next chapter (Section 6.2.5) — consider moving them here aftersome editing!

Lemma. If b ∈ B is a diagram with an odd number of legs, all of whichare attached to one and the same edge, then b = 0 modulo AS and IHXrelations.

Conjecture. Any diagram with an odd number of legs is 0 in B.

5.7. Linear isomorphism B ≃ CIn this section we construct a linear isomorphism between vector spaces Bnand Cn. The question whether it preserves multiplication will be discussedlater on (section 5.8). Our exposition follows [BN1], with some detailsomitted, but some examples added.

To convert an open diagram into a closed diagram, we will join all ofits univalent vertices (legs) by a Wilson loop. For an open diagram with klegs D ∈ ODk

n there are k! ways1 of doing that, and we set χ(D) to be equalto the arithmetic mean of all the resulting closed diagrams. Thus we get amapping

χ : OD→ C.

1Well, actually (k − 1)!, not k!, because the external vertices of a closed diagram have a

cyclic, rather than linear, ordering. But it is somewhat easier to enumerate all linear orderings.


For example,

χ( )

=1

24

(+ + + + +

+ + + + + +

+ + + + + +

+ + + + + +

).

Scrutinizing these pictures, one can see that 16 out of 24 summands areequivalent to the first diagram, while the remaining 8 are equivalent to thesecond one. Therefore,

χ( ) =1

3+

2

3.

Exercise Express this element through chord diagrams, using the iso-morphism C ≃ A.

Answer: − 10

3+

4

3.

5.7.1. Theorem. The mapping χ : OD → C descends to a linear operatorχ : B → C which is a graded isomorphism between the linear spaces B andC.

The theorem consists of two parts:

• Easy part: the linear mapping χ is well-defined.

• Difficult part: the mapping χ is bijective.

Easy part of the theorem. To prove the easy part, we must showthat the AS and IHX combinations of open diagrams go into 0 in the spaceC. This follows from lemmas 5.2.5 and 5.2.6.

Difficult part of the theorem. To prove the difficult part, we willconstruct a linear mapping τ from C to B, inverse to χ.

As usual, we first define this mapping on generators. If an elementC ∈ CD is symmetric, that is C = χ(B) for a certain combination of opendiagrams B ∈ B, then, of course, we can just set τ(C) = B.


For example, the element is not symmetric (although it looks

so), because symmetrizing it over all permutations of its external vertices,

we get 0. On the other hand, the element is symmetric, as it is equal

to the image of the open diagram under χ. This is true about anyclosed diagram with just two legs, because on the set of two elements thereis only one cyclic order (in other words, one leg of the diagram can slideover the Wilson loop and thus change places with another). Therefore, theidea of constructing the mapping τ inductively by the number of externalvertices naturally arises.

Note that the difference of any closed diagram and the same diagramwhose legs are permuted by an arbitrary permutation, is equivalent, moduloSTU relations, to a combination of diagrams with a smaller number of ex-ternal vertices. Assuming that the mapping τ is defined for closed diagramshaving less than k legs, we define it for a diagram D with exactly k legs bythe formula:

(1) τk(D) = D +1

k!

∑

π∈Sk

τk−1(D − π(D)) ,

where D is the open diagram that you get by stripping the Wilson loopoff D and π(D) denotes the action of the symmetric group Sk on closeddiagrams by permutations of external vertices (of course, the action requiresa numbering of these vertices, but the sum does not depend on a particularnumbering).

For example, we know that τ( )

= , and we want to find

τ( )

. By the above formula, we have:

τ3

( )= + 1

6

(τ2

(−

)+ τ2

(−

)

+τ2

(−

)+ τ2

(−

)

+τ2

(−

)+ τ2

(−

))

= 12τ2

( )= 1

2 .


We have to prove the following assertions:

(i) the mapping τ is well-defined, i.e. the value τk−1(D − π(D)) inthe formula (1) does not depend on the presentation of D − π(D)as a combination of diagrams with a smaller number of externalvertices.

(ii) the mapping τ satisfies STU relations and thus descends to a map-ping τ : C → B.

(iii) χ τ = idC .

In the next subsections we complete these tasks following D. Bar-Natanpaper [BN1] with minor improvements.

Let Dk be the subspace of 〈CD〉 spanned by all closed diagrams with atmost k external vertices. We have an increasing filtration

D0 ⊂ D1 ⊂ D2 ⊂ · · · ⊂ D := 〈CD〉 .

Denote by IDk be the subspace in Dk spanned by all STU, IHX and anti-symmetry relations that do not involve diagrams with more than k exter-nal vertices. Let β : Dk−1 → Dk/Ik be the composition of the inclusionDk−1 → Dk and the projection Dk → Dk/Ik.

5.7.2. Action of permutations on closed diagrams. The permutationgroup Sk acts on diagrams in Dk having precisely k external vertices bypermuting those vertices (provided that these vertices are numbered). Thisaction can be represented graphically as the composition of a closed diagramwith the diagram of the permutation:

k = 4 ; π = (4132) = ; D = ;

πD = = .

5.7.3. Lemma.

• Regarded as an element of Dk/Ik, the difference D − πD belongsto the image of β.

• Any choice Uπ of a presentation of π as a product of transpositionsdetermines in a natural way an element ΓD(Uπ) ∈ Dk−1 for whichβ(ΓD(Uπ)) = D − πD.

• Furthermore, if Uπ and U ′π are two such presentations, then ΓD(Uπ)is equal to ΓD(U ′π) modulo Ik−1.


This is Lemma 5.5 from [BN1]. Referring the interested reader to [BN1]for alldetails, we confine ourselves with an illustration which visually conveysthe idea of the proof. Take the permutation π = (4132) and let D be thediagram considered above. Choose two presentations of π as a product oftranspositions:

Uπ = (34)(23)(34)(12) = ; U ′π = (23)(34)(23)(12) = .

For each of them we represent D − πD as a corresponding telescopic sum:

D − πD = (D − (12)D) + ((12)D − (34)(12)D) + ((34)(12)D − (23)(34)(12)D)+((23)(34)(12)D − (34)(23)(34)(12)D)

and

D − πD = (D − (12)D) + ((12)D − (23)(12)D) + ((23)(12)D − (34)(23)(12)D)+((34)(23)(12)D − (23)(34)(23)(12)D) .

Here, the two terms in every pair of parentheses differ only by a transpositionof two neighboring legs, so their difference represents the right hand side ofan STU relation. Modulo the subspace Ik each difference can be replacedby the corresponding left hand side of the STU relation wich is a diagramin Dk−1. We get

ΓD(Uπ) = + + +

ΓD(U ′π) = + + +

Now the difference ΓD(Uπ)− ΓD(U ′π) equals

(−

)+(

−)

+(

−)

Using the STU relation in I3 we can represnt it in the form

ΓD(Uπ)− ΓD(U ′π) = + − = 0

which is zero because of the IHX relation.


5.7.4. Definition of the mapping τ : D → B. We will inductively con-struct a sequence of compatible maps τk : Dk → B satisfying

(Σ1) τk(Ik) = 0;(Σ2) τk|Dk−1 = τk−1 .

And then put τ = τ∞.

Case k = 1. A diagram D ∈ D1 has exactly one vertex on its circle.Define τ1(D) to be D with its Wilson loop removed and then (Σ1) is trivialbecause the AS and IHX relation hold in B as well and there is no STUrelation in D1.

Let us assume for some k > 1 a map τk−1 satisfying (Σ1) and (Σ2) wasconstructed.

Lemma 5.7.3 means that for any given D ∈ Dk with exactly k externalvertices τk−1(ΓD(Uπ))) does not depend on a specific presentation Uπ of apermutation π as a product of transpositions. So we can define τk(D) forsuch a diagram D according to the formula (1):

τk(D) = D +1

k!

∑

π∈Sk

τk−1(ΓD(Uπ))) .

For a diagram D having less than k external vertices, just set τk(D) =τk−1(D), according to (Σ2).

This definitions gives a well defined linear map Dk → B. We only haveto check the property (Σ1) for it. It is obvious that IHX and antisymmetryrelations transform by τk to zero because there are the same relations in B.So we need only to check the STU relation which relates a diagram Dk−1

with k − 1 external vertices and corresponding two diagrams Dk and UiDk

with k external verices, where Ui is a transposition Ui = (i, i+ 1). By (Σ2)τk(Dk−1) = τk−1(Dk−1). Now let us apply τk to the right hand side of theSTU relation:

τk(Dk − UiDk) = Dk + 1k!

∑π∈Sk

τk−1(ΓDk(Uπ))

−UiDk − 1k!

∑π′∈Sk

τk−1(ΓUiDk(Uπ′)) .

Note that Dk = UiDk. Reparametrizing the first sum, we get

τk(Dk − UiDk) =1

k!

∑

π∈Sk

τk−1(ΓDk(UπUi)− ΓUiDk(Uπ)) .

5.8. Relation between A and B 127

Using an obvious identity ΓD(UπUi) = ΓD(Ui) + ΓUiDk(Uπ) and the fact

that Dk−1 = ΓD(Ui), we now obtain

τk(Dk − UiDk) =1

k!

∑

π∈Sk

τk−1(Dk−1) = τk−1(Dk−1) ,

which means that τk satisfies the STU relation.

5.7.5. Proof of assertion (ii). Now that the map τ is constructed, theequality τ(I∞) = 0 follows from (Σ1). Therefore the map τ : D → Bdescends to a map τ : C → B

It is easy to see from the definition that τ is surjective, so τ is alsosurjective. The injectivity of τ follows from invertibility proved in the nextsection.

5.7.6. Proof of assertion (iii). Let D ∈ Dk be regarded modulo Ik.Then

(χ τk)(D) = χ(D + 1

k!

∑π∈Sk

τk−1(ΓD(Uπ)))

= 1k!

∑π∈Sk

(πD + (χ τk−1)(ΓD(Uπ))

).

By the induction hypothesis χ τk−1 = idC . Hence

(χ τk)(D) =1

k!

∑

π∈Sk

(πD + ΓD(Uπ)

)=

1

k!

∑

π∈Sk

(πD +D − πD

)= D .

This concludes the proof of the linear isomorphism of C and B.

5.8. Relation between A and BIt is easy to check that the isomorphism χ is compatible with the coproductin the algebras A and B. (Exercise: pick a decomposable diagram b ∈ Band compare δA(χ(b) with χ(δB(b)) to make sure these elements literallycoincide.) However, χ is not compatible with the product. For example,

χ( ) = .

The square of the element in B is . However, the corre-sponding element of C

χ( ) =1

3+

2

3


is not equal to the square of .

We can, of course, carry the natural multiplication of the algebra C ontothe algebra B along the isomorphism χ, thus obtaining a Hopf algebra withtwo different products, both compatible with one and the same coproduct— a very interesting algebraic object indeed.

Although the isomorphism χ does not take one multiplication into an-other, the two algebras A and B are still isomorphic. This is clear from whatwe know about their structure: both algebras satisfy the Milnor–Moore the-orem, so both are commutative polynomial algebras over the correspondingprimitive subspaces. But the primitive subspaces coincide, because χ pre-serves the coproduct! An explicit algebra isomorphism between A and Bwill be the subject of Section 10.5.

By definition, any connected diagram p ∈ B is primitive. An analog ofTheorem 5.5.1 holds.

5.8.1. Theorem. The primitive space of the bialgebra B is spanned by con-nected open diagrams.

Proof. The same argument as in the proof of Theorem 5.5.1, with a simpli-fication that in the present case we do not have to prove that every elementof B has a polynomial expression in terms of connected diagrams: this is soby definition.

Warning : the assertion of the theorem is not evident, because in theexpansion of δ(x) where x ∈ B is a linear combination of connected diagrams,all the terms except 1 ⊗ x + x ⊗ 1, might cancel each other — in fact thiscannot happen, because the theorem is proven!

5.8.2. Grading in B and filtration in A. The coproduct in A corre-sponds to the coproduct in B via the isomorphism χ. Therefore, the corre-sponding primitive spaces PA and PB are carried by χ one into another.The space PA is filtered (section 5.5.2), the space PB is graded (page 120).It turns out that isomorphism χ intertwines this filtration with this grading.Indeed, the definition of χ and he construction of the inverse mapping τimply two facts:

χ(PBi) ⊂ PAi ⊂ PAk, if i < k,

τ(PAk) ⊂k⊕

i=1

PBk.

5.9. The three algebras in small degrees 129

We have thus an isomorphism

τ : Pkn −→ PB1n ⊕ PB2

n ⊕ . . .⊕ PBk−1n ⊕ PBkn .

and therefore an isomormphism Pkn/Pk−1n∼= PBkn.

Using this fact, we can give an elegant reformulation of the theoremabout detecting the orientation of knots (Section 5.5.3, p.117):

Corollary Vassiliev invariants do distinguish the orientation of knots ifand only if PBkn 6= 0 for an odd k and some n.

To avoid misunderstanding, let us mention that the phrase Vassilievinvariants do distinguish the orientation of knots has the following precisemeaning: there exists a knot K non-equivalent to its inverse K∗ and a Vas-siliev invariant f such that f(K) 6= f(K∗).

5.9. The three algebras in small degrees

Here is a comparative table which displays some linear bases of the algebrasA, C and B in small dimensions.

n A C B

0 ∅

1

2

3

4

In every grading up to 4, for each of the three algebras, this table displaysa basis of the corresponding homogeneous component. Starting from grading2, decomposable elements (products of elements of smaller degree) appearon the left, while the new indecomposable elements appear on the right.The bases of C and B are chosen to consist of primitive elements and theirproducts. We remind that the difference between the A and C columnsis rather conditional, because chord diagrams are a special case of closed


Jacobi diagrams, the latter can be considered as linear combinations of theformer, and the two algebras are in any case isomorphic.

Exercises

(1) Let a1 = , a2 = , a3 = , a4 = , a5 = .

a). Find a relation between a1 and a2.b). Represent the sum a3 + a4 − 2a5 as a connected closed diagram.c). Prove the linear independence of a3 and a4 in C.

(2) Express the basic primitive elements , , and of degrees

3 and 4 as linear combibations of chord diagrams.

(3) Show that the symbols of the coefficents of the Conway polynomial (Sec.2.3) take the following values on the basic primitive diagrams of degree3 and 4.

symb(c3)( )

= 0,

symb(c4)( )

= 0, symb(c4)( )

= −2.

(4) Show that the symbols of the coefficents of the Jones polynomial (Sec.3.7.2) take the following values on the basec primitive diagrams of de-grees 3 and 4.

symb(j3)( )

= −24,

symb(j4)( )

= 96, symb(j4)( )

= 18.

(5) ([CV]) Let tn ∈ Pn+1 be a closed diagramshown on the right figure. Prove the followingidentity

tn =1

2n

n bubbles

tn =

n legs

Deduce that tn ∈ P2n+1.

Exercises 131

(6) Express tn as a linear combination of chord diagrams. In particular,show that the intersection graph of every chord diagram occurred in theexpression ia a forest.

(7) ([CV]) Prove the following identity in the space C of closed diagrams:

=3

4− 1

12− 1

48.

Hint. Turn the internal pentagon of the right hand side of the iden-tity as a rigid thing in the 3-space by 180 about the vertical axis. Theresult will represent the same graph only the cyclic orders at all fivevertices of the pentagon will be changed to the opposite:

= (−1)5 = − + (terms with at most 4 legs) .

The last equality follows from the STU relations which allow us to re-arrange the legs modulo diagrams with a smaller number of legs. Tofinish the solution, the reader must figure out what are the terms inparentheses.

(8) Prove the linear independence of the three elements in the right handside of the last equality, using Lie algebra invariants to be defined later,in Chapter 7.

(9) ([CV]) For a closed diagram C, let C \ W denotes a graph obtainedfrom C by erasing the Wilson loop. Prove that the primitive space P isgenerated by those closed diagrams C for which C \W is a tree.

(10) ([CV]) With each permutation π of n objects associate a closed dia-gram Pπ acting by the permutation on the lower legs of a closed dia-gram P(12...n) = tn from problem 5 like in section 5.7.2. Here are a fewexamples:

P(2143) = ; P(4123) = ; P(4132) = .

Prove that diagrams Pπ span the vector spase Pn+1.

(11) ([CV])•• ([CV]) Prove that Pnn = Pn for even n, and Pn−1

n = Pn for odd n.• Prove that for even n the quotient space Pnn/Pn−1

n has dimensionone and generated by the wheel wn.


(12) Let b1 = , b2 = , b3 = , b4 = .

Which of these diagrams are zero in B, i.e. vanish modulo AS and IHXrelations?

(13) The lemma in Section 5.5.3 and the isomorphism χ : A → B implythat an open Jacobi diagram with only one leg is 0 in B. Find a directindependent proof of this fact.

(14) What will happen if, in the construction of B, the IHX relation is re-placed by the following relation: I = εH − ε2X ?

(15) • Find some bases of the spaces An, Cn, Bn for n = 4.• Indicate an explicit form of the isomorphisms A ∼= C ∼= B in these

bases.• Compile the multiplication table for Bm×Bn → Bm+n, m+ n ≤ 4,

for the second product in B (the one pulled back from C along theisomorphism C ∼= B).

(16) (J. Kneissler). Let Bun be the space of open diagrams of degree n with

u univalent vertices. Denote by ωi1i2...ik the element of Bi1+···+iki1+···+ik+k−1

represented by a “caterpillar” diagram consisting of k body segmentswith i1, . . . , ik “legs”, respectively. Using the AS and IHX relations,prove that ωi1i2...ik is well-defined, i.e., for inner segments it makes nodifference on which side of the body they are drawn). For example,

ω0321 = =

(17) ∗ (J. Kneissler) Is it true that any caterpillar diagram in the algebra Bcan be expressed through caterpillar diagrams with even indices i1, . . . ,ik? Is it true that the primitive space P(B) (i.e. the space spanned byconnected open diagrams) is generated by caterpillar diagrams?

(18) Call a chord diagram d ∈ A even (odd) if d = d (resp. d = −d), wherethe bar denotes the reflection of a chord diagram. Prove that under theisomorphism χ−1 : A → B:• the image of an even chord diagram is a linear combination of open

diagrams with an even number of legs,• the image of an odd chord diagram in is a linear combination of

open diagrams with an odd number of legs.

(19) ∗ Is it true that an open diagram with 3 univalent vertices is alwaysequal to 0 as an element of the algebra B?

Chapter 6

Algebra of 3-graphs

The algebra of 3-graphs, introduced in [DKC], is obtained by a construc-tion which is very natural from an abstract point of view. The elements ofthis algebra differ from closed diagrams in that they do not have any distin-guished cycle, like Wilson loop; they differ from open diagrams in that thesegraphs are regular, without univalent vertices. So it looks as a simplificationof both algebras C and B. Strictly speaking, there are two different algebrastructures on the same linear space Γ of 3-graphs given by the edge (section6.2) and the vertex (section 6.3) multiplications. These algebras are closelyrelated to the Vassiliev invariants in several ways. Namely,

• the vector space Γ is isomorphic to the subspace P2 of the primitivespace P of closed diagrams generated by connected closed diagramswith 2 legs (section 6.4.2);

• the algebra Γ acts on the primitive space P in two ways, via edge,and via vertex multiplications (see sections 6.4.1 and 6.4.3);

• these actions behave nicely with respect to Lie algebra weight sys-tems (see chapter 7); as a consequence, the algebra Γ is as good atool for the proof of existence of non-Lie-algebraic weight systemsas the algebra Λ in Vogel’s original approach (section 7.5);

• and finally, the algebra Γ describes the combinatorics of finite typeinvariants of integral homology 3-spheres precisely in the same wayas the algebra of chord diagrams describes the combinatorics ofVassiliev knot invariants. This, however, lies outside of the scopeof this book and we refer an interested reader to [Oht1].

Unlike A and B, the algebra Γ does not have any natural coproduct.

133

134 6. Algebra of 3-graphs

6.1. The space of 3-graphs

6.1.1. Definition. A 3-graph is a connected 3-valent graph1 with a fixedrotation. The rotation is the choice of a cyclic order of edges at every vertex,i. e. one of the two cyclic permutations in the set of three edges adjacent tothis vertex.

In this definition graphs are allowed to have multiple edges and loops.In the case of graphs with loops the notion of rotation requires the followingrefinement: the cyclic order is introduced not in the set of edges incident toa given vertex, but in the corresponding set of half-edges.

Remark. Graphs with rotation are often called ribbon graphs, becausesuch a graph can be represented as an orientable surface with boundaryhaving the form of a narrow strip along the skeleton that has the shape ofthe given graph:

A topology free combinatorial definition of the set of half-edges can begiven as follows. Let E, V be the sets of edges and vertices, respectively,of the graph under study. Then the set of half-edges is a ‘double fibering’

Eα← H

β→ V , that is a set H supplied with two projections α : H → E andβ : H → V such that (a) the inverse image of every edge α−1(e) consistsof two elements, (b) the cardinality of each set β−1(v) equals the valencyof the vertex v, (c) for any point h ∈ H the vertex β(h) is one of the twoendpoints of the edge α(h). Then the rotation is a cyclic permutation in theinverse image of each vertex β−1(v).

One can forget about edges and vertices altogether and adopt the follow-ing clear-cut, although less pictorial, definition of a 3-graph: this is a set ofcardinality 6n equipped with two permutations, one with the cyclic structure(3)(3) . . . (3), another with cyclic structure (2)(2) . . . (2), with the require-ment that there is no proper subset invariant under both permutations. Therelation to the previous definition consists in that H is the set of half-edges,the first permutation corresponds to the rotation in the vertices and thesecond, to the transition from one endpoint of an edge to another.

6.1.2. Definition. Two 3-graphs are said to be isomorphic if there is aone-to-one correspondence between their sets of half-edges which preserves

1Note that the number of vertices of such a graph is always even, and the number of edges

a multiple of 3. Half the number of the vertices will be referred to as the degree (or order) of a

3-graph.

6.2. Edge multiplication 135

the rotation and induces a usual graph isomorphism — or, in other words,which preserves both structural permutations.

Remark. It is convenient to include the circle into the set of 3-graphsviewing it as a graph with a zero number of vertices.

Example. Up to an isomorphism, there are three different 3-graphs ofdegree 1:

6.1.3. Definition. The space Γn is the quotient space of the vector spaceover Q spanned by connected 3-graphs of degree n (i. e. having 2n vertices),modulo the AS and IHX relations (see p. 108). By definition, Γ0 is one-dimensional and generated by the circle.

That is, the space of 3-graphs differs from the space of open diagrams(p. 119) in that here we consider only connected diagrams without univalentvertices, while the space B is spanned by diagrams with necessarily at leastone univalent vertex in each connected component.

6.1.4. Exercise.Check that the graph on the right is equal to zero as anelement of the space Γ3.

6.2. Edge multiplication

In the graded space

Γ = Γ0 ⊕ Γ1 ⊕ Γ2 ⊕ Γ3 ⊕ . . . .there is a natural structure of a commutative algebra.

6.2.1. Definition. The edge product of two non-zero 3-graphs x and y isthe 3-graph obtained in the following way. Choose arbitrarily an edge in xand an edge in y. Cut each of these two edges in the middle and attachthe endpoints that appear on one edge, to the endpoints appearing on theother.

x

y

x

y

= x · y .

The edge product of 3-graphs can be understood as the connected sumof x and y along the chosen edges, and also as the result of insertion of onegraph, say x, into an edge of y.


Remark. The product of two connected graphs may yield a disconnectedgraph, for example:

× =

This happens, however, only in the case when each of the two graphs be-comes disconnected after cutting the chosen edge, and in this case bothgraphs are 0 modulo AS and IHX relations (see Lemma 6.2.7(b)).

6.2.2. Theorem. The edge product of 3-graphs, viewed as an element ofthe space Γ, is well-defined.

Note that, as soon as this assertion is proved one immediately sees thatthe multiplication is commutative.

The claim that the product is well-defined consists of two parts. First,that modulo the AS and IHX relations the product does not depend on thechoice of the two edges x and y which are cut and pasted. Second, that itdoes not depend on the way they are put together (clearly, the two looseends of one graph can be glued to the two loose ends of another graph intwo different ways). These two facts are established in the following twolemmas.

6.2.3. Lemma.(a) A vertex with an attached edge canbe dragged through a subgraph that hastwo legs.

x

y′=

x

y′

(b) A subgraph with two legs can becarried through a vertex.

x=

x

Proof. Taking a closer look at these pictures, the reader will understandthat assertions (a) and (b) have exactly the same meaning and in fact arenothing but a particular case (k = 1) of the generalized Kirchhoff law (seepage 111).

Another wording for Lemma 6.2.3 is: the results of insertion of a 3-graphx into the adjacent edges of the 3-graph y, are equal. Since y is connected,this implies that the product does not depend on the choice of the edge iny.


6.2.4. Lemma. Two different ways to glue together two graphs with edgescut give the same result in the space Γ:

x

y=

x

y.

Proof. Choosing the vertex of x which is nearest to the right exit of y, onecan, by Lemma 6.2.3, effectuate the following manoeuvres:

x

y

= xy

= x

y

= x

y

.

Therefore,x

y=

x

y

=

x

y.

The lemma is proved, and edge multiplication of 3-graphs is thus well de-fined.

The edge product of 3-graphs is extended to the product of arbitraryelements of the space Γ in the usual way, by linearity.

Corollary. The edge product in Γ is well-defined, associative, and dis-tributive.

This follows from the fact that the product of 3-graphs is well-definedand a linear combination of either AS or IHX type, when multiplied byan arbitrary graph, is a combination of the same type. Distributivity andassociativity are obvious.

6.2.5. Some identities. We will prove several identities elucidating thestructure of the 3-graph algebra. They are easy consequences of the definingrelations. Note that these identities make sense and hold also in the algebrasof closed and open diagrams, C and B.

The first identity (Lemma 6.2.6) links the two operations defined in theset of 3-graphs: the insertion of a triangle into a vertex (which is a particularcase of the vertex multiplication we shall study later) and the insertion of abubble into an edge. To insert a bubble into an edge of a graph is the samething as to multiply this graph by the element β = iq q∈ Γ1. The correctnessof multiplication in Γ implies that the result of the bubble insertion doesnot depend on the edge chosen, then it follows from Lemma 6.2.6 that theresult of the triangle insertion does not depend on the choice of the vertex.


6.2.6. Lemma. A triangle is equal to one half of a bubble:

=1

2=

1

2=

1

2.

Proof.

= + = − .

The second lemma describes two classes of 3-graphs which are equal to0 in the algebra Γ, i. e. modulo the AS and IHX relations.

6.2.7. Lemma.(a) A graph with a loop is 0 in Γ. = 0

(b) More general, if the edge connectivityof the graph γ is 1, i. e. it becomes discon-nected after the removal of an edge, thenγ = 0.

γ = = 0

Proof. (a) A graph with a loop is zero because of antisymmetry. Indeed,changing the rotation in the vertex of the loop yields a graph which is, onone hand, isomorphic to the initial one, and on the other hand, differs fromit by a sign.

(b) Such a graph can be represented as a product of two graphs, one ofwhich is a graph with a loop, i.e. a zero according to (a):

γ = x y = x × y = 0 .

6.2.8. The Zoo. The next table shows the dimensions dn and displays thebases of the vector spaces Γn for n ≤ 11, obtained by computer calculations.

From the table one can see that the following elements can be chosen asmultiplicative generators of the algebra Γ up to degree 11 (notations in thelower line mean: β — ‘bubble’, ωi — ‘wheels’, δ — ‘dodecahedron’):

1 4 6 7 8 9 10 11

β ω4 ω6 ω7 ω8 ω9 ω10 δ ω11


6.2.9. Conjecture. The algebra Γ is generated by planar graphs.

The reader might have noted that the table of additive generators doesnot contain the elements ω2

4 of degree 8 and ω4ω6 of degree 10. This is notaccidental. It turns out that the following relations (found by A. Kaishev[Kai]) hold in the algebra Γ:

ω24 =

5

384β8 − 5

12β4ω4 +

5

2β2ω6 −

3

2βω7,

ω4ω6 =305

27648β10 − 293

864β6ω4 +

145

72β4ω6 −

31

12β3ω7 + 2β2ω8 −

3

4βω9.

In fact, as we shall see in section 6.3, it is true in general that the productof an arbitrary pair of homogeneous elements of Γ of positive degree belongsto the ideal generated by β.

Since there are non-trivial relations between the generators, the algebraof 3-graphs, in contrast to the algebras A, B and C, is not free and hencedoes not possess the structure of a Hopf algebra.


n dn additive generators

1 1

2 1

3 1

4 2

5 2

6 3

7 4

8 5

9 6

10 8

11 9

6.2.10. Conjecture. The mapping Γ → Γ of edge multiplication by thebubble (x 7→ β · x) has no kernel.

This is an old conjecture: in the context of Vassiliev invariants, it wasstated several times by various mathematicians, beginning from 1992, andstood several unsuccessful attempts of proof. We generalize it as follows.

6.2.11. Conjecture. The algebra Γ has no zero divisors.

6.3. Vertex multiplication 141

6.3. Vertex multiplication

In this section we define the vertex multiplication on the space

Γ≥1 = Γ1 ⊕ Γ2 ⊕ Γ3 ⊕ Γ4 ⊕ . . . .It is different from Γ that it does not contain the one dimensional subspaceΓ0 generated by the circle, the only graph without vertices.

6.3.1. Definition. The vertex product of two 3-graphs G1 and G2 is the3-graph obtained in the following way. Choose arbitrarily a vertex in G1 anda vertex in G2. Cut each of these two vertices together with the half-edgesincident to them. Then attach the endpoints that appear on one graph, tothe endpoints appearing on the other. There are six possibilities for this.Take the alternating average of all of them. We take with the sign minusthose three attachments where the cyclic orders at two removed vertices arecoherent and with the plus sign those three which switch the cyclic orders.

Pictorially, if we draw the graphs G1, G2 like this G1 =G1

, G2 =G2

,

then to draw their vertex product we have to merge them and insert apermutation of the three strands between them. Then we take the resultwith the sign of the permutation and average it over all six permutations:

G1 ∨G2

=1

6

[

G2

G1

−G2

G1

−G2

G1

−G2

G1

+G2

G1

+G2

G1].

As an example, let us compute the vertex product by a bubble β:

β ∨G

= ∨G

=1

6

[

G−

G−

G−

G+

G+

G

]=

G,

because all summands in the brackets are equal to each other since the ASrelation for the vertex. So β is going to be the unit for the algebra Γ≥1 withvertex multiplication.

The vertex product of 3-graphs can be understood as the average resultof insertions of one graph, say G1, into a vertex of G2 over all possible ways.

To shorten the notation we will draw diagrams with shaded disks, un-derstand such a diagram as an alternating linear combination of six graphslike above. For example:

G1 ∨G2

=G2

G1

=G2

G1

.

6.3.2. Theorem. The vertex multiplication in Γ is well-defined, commuta-tive and associative.


Proof. We must only prove that the following equality holds due to the ASand IHX relations:

X1 = G = G = X2.

where G denotes an arbitrary element of Λ (see 6.5).

By the generalized Kirchhoff law we have:

*G =

*G +

*

G

= G + G + G + G

(the stars indicate the place where the tail of the ‘moving electron’ is fixed inKirchhoff’s relation). Now, in the last line the first and the fourth diagramsare equal to X2, while the sum of the second and the third diagrams is equalto −X1 (again, by an application of generalized Kirchhoff’s rule). We thushave 2X1 = 2X2 and therefore X1 = X2.

Commutativity and associativity are obvious.

6.3.3. Remark. The vertex multiplication shifts the degrees by one. So itis an operation of degree −1: Γn ∨ Γm ⊂ Γn+m−1.

6.3.4. Proposition. (Relation between two multiplications in Γ.)The edge multiplication “·” in the algebra of 3-graphs Γ is related to thevertex multiplication “∨” as follows:

G1 ·G2 = β · (G1 ∨G2)

Proof. Choose a vertex in each of the given graphs G1 and G2 and callwhat remains G′1 and G′2, respectively:

G1 =G′

1

=G′

1

, G2 =G′

2

,

where, as explained above, the shaded region means an alternating averageover the six permutations of the three legs.

6.4. Action of Γ on the primitive space P 143

Then, by theorem 6.3.2 we have:

G1 ·G2 =G′

1

G1

G′2

G2

=G′

1 G′2

= β · G′1 G′

2= β · (G1 ∨G2) .

Corollary. The algebra (Γ, ·) is not a Hopf algebra.

6.4. Action of Γ on the primitive space P6.4.1. Edge action of Γ on P. As we know (section 5.5) the primitivespace P of the algebra C is generated by connected closed diagrams, i. e.those diagrams which remain connected after the Wilson loop is strippedoff. It is natural to define the edge action of Γ on such a diagram P simplyby edge multiplication of a graph G ∈ Γ with P , using an internal edge ofP . More specifically, we cut an arbitrary edge of the graph G. Then we cutan edge in P not lying on the Wilson loop and attach the new endpoints ofP to those of G, like we did before for the algebra Γ itself. The graph thatresults is G · P ; it has one Wilson loop and is connected, hence it belongsto P. Because of the connectivity of P and lemmas 6.2.3, 6.2.4 this resultG · P does not depend on the choices of the edge in G and the internaledge in P . So this action is well-defined. The action is compatiblewith thegradings: Γn · Pm ⊂ Pn+m. We shall use the edge action in the nextchapter in conjunction with Lie-algebraic weight systems.

6.4.2. Proposition. The vector space Γ = Γ0 ⊕ Γ1 ⊕ Γ2 ⊕ Γ3 ⊕ . . . isisomorphic to the subspace P2 = P2

1 ⊕P22 ⊕P2

3 ⊕P24 ⊕· · · ⊂ P of closed dia-

grams generated by connected diagrams with 2 legs as graded vector spaces:Gn ∼= Pn+1 for n ≥ 0.

Proof. The required mapping Γ → P2 is given by the edge action of 3-graphs on the element Θ ∈ P2 represented by the chord diagram with asingle chord, G 7→ G · Θ. The inverse mapping can be constructed in anatural way. For a closed diagram P ∈ P2 strip off the Wilson loop andglue together the two loose ends of the obtained diagram. The result will bea 3-graph of degree one less than P . Obviously this mapping is well-definedand inverse to the previous one.


6.4.3. Vertex action of Γ on P. To perform the vertex multiplication weneed at lest one vertex in each of the factors. So the natural action we arespeaking about is the action of the algebra Γ≥1 (with the vertex product)on the space P>1 of primitive elements of degree strictly greater than 1.The action is defined in a natural way. Pick up a vertex in G ∈ Γ≥1, andinternal vertex in P ∈ P>1. Then the result G ∨ P will be the alternatingaverage over all six ways to insert G with the removed vertex into P withthe removed internal vertex. Again because of connectivity of P and theTheorem 6.3.2 the action is well-defined. This action decreases the totalgrading by 1 and preserves the number of legs: Γn ∨ Pkm ⊂ Pkn+m−1.

The simplest element of P on which one can act in this way is the “Mer-

cedes closed diagram” t1 = . This action has the following important

property.

6.4.4. Lemma. (a) The map Γ≥1 → P defined as G 7→ G ∨ t1 is an inclu-sion, i. e. it has no kernel.

(b) The vertex and edge actions are related to each other via the formula:

G ∨ t1 =1

2G ·Θ .

This lemma will be used in the next chapter (section 7.5) for Lie algebraweight systems.

Proof. Indeed, t1 = 12β ·Θ. Therefore G∨ t1 = 1

2(G∨β) ·Θ = 12G ·Θ. Since

the mapping G 7→ G ·Θ is an isomorphism (proposition 6.4.2), the mappingG 7→ G ∨ t1 is also an isomorphism Γ≥1

∼= P2>1.

6.4.5. Other relations between Γ and P. There are additional funnyrelations between Γ and P which we are going to describe now.

By construction, the space of primitive elements P of the algebra C(see Sec. 5.5), considered by itself, does not possess any a priori definedmultiplicative structure. Primitive elements only generate the algebra C inthe same way as the variables x1, . . . , xn generate the polynomial algebraR[x1, . . . , xn]. However, the link between the space P and the algebra of3-graphs Γ allows one to introduce in P a natural (non-commutative) mul-tiplication. We shall now describe how this is done.

As we know (see section 5.5), the primitive space of the algebra C coin-cides with the subspace spanned by the connected closed diagrams.

The relation between the vector spaces Γ and P is expressed by twofacts:

6.5. Vogel’s algebra Λ 145

1. There is a linear inclusion i : Γn → Pn+1, defined on the generatorsas follows: we cut an arbitrary edge of the graph and attach a Wilson loopto the two resulting endpoints. The orientation of the Wilson loop can bechosen arbitrarily. This is precisely the inclusion from proposition 6.4.2:G 7→ G ·Θ.

2. There is a projection π : Pn → Γn, which consists in introducing therotation in the vertices of the Wilson loop according to the rule ‘forward–sideways–backwards’ and then forgetting the fact that the Wilson loop wasdistinguished.

The composition homomorphism Γ→ Γ in the sequence

Γi−→ P π−→ Γ

coincides with the multiplication by the bubble G 7→ β ·G.

The edge action · : Γ×P → P gives rise to an operation ∗ : P ×P → Pwhich can be defined by the rule

p ∗ q = π(p) · q,where π : P → Γ is the homomorphism of forgetting the Wilson loop definedabove.

The operation ∗ is, in general, non-commutative:

∗ = , but ∗ = .

These two elements of the space P are different; they can be distin-guished, e. g., by the sl2-invariant (see exercise 22 at the end of the nextchapter). However, π projects these two elements into the same elementβ · ω4 ∈ Γ5.

6.4.6. Conjecture. The algebra of primitive elements P has no divisors ofzero with respect to multiplication ∗.

6.5. Vogel’s algebra Λ

In this section, we descrive the relation between the algebra Γ and Vogel’salgebra Λ.

Diagrams with 1- and 3-valent vertices can be considered with differentadditional structures on the set of univalent vertices (legs). If there is nostructure, then we get the notion of an open Jacobi diagram. If the legsare attached to a circle or a line, then we obtain closed Jacobi diagrams. Adiagram with a linear order (numbering) on the set of legs, but without STUrelations, will be called a fixed diagram. By definition, two such diagrams


are isomorphic, if between them there is an isomorphism that preserves theleg numbering.

We will denote the set of all such diagrams equipped, as usual, with arotation in the 3-valent vertices, by Y . By definition, speaking about thenumbering means that an isomorphism between two such diagrams mustpreserve the numbering of the legs. In the set Y there are two gradings:by the number of legs (denoted by the superscript) and by half the totalnumber of vertices (denoted by the subscript).

6.5.1. Definition. The vector space spanned by connected fixed diagramswith k legs modulo the usual AS and IHX relations

X kn = 〈Y kn 〉/〈AS, IHX〉,

is called the space of fixed diagrams of degree n with k legs. (Everything isover a field of characteristic 0.)

Remark. There are a lot of relations between the spaces X k. Forexample, one may think about the diagram as a linear operator from X 4

to X 3. Namely, it acts on an element G of X 4 as follows

: G

123 4

7→G

1 23

.

Exercise. Prove the following relation

+ + = 0,

between the three linear operators from X 4 to X 3.

The space of open diagrams Bk studied in Chapter 5 is the quotient ofX k by the permutation group Sk that renumbers the legs. The mapping oftaking the quotient X → B has a nontrivial kernel, for example, a tripodwhich is nonzero in X 3 becomes zero in B:

0 6=21

3

7−→ = 0.

6.5.2. Definition. The algebra Λ is the subspace of all elements of X 3, an-tisymmetric with respect to the permutations of legs. The multiplication isdefined on the generators via a sort of vertex multiplication. Namely, choosea vertex in the first diagram, remove it together with 3 adjacent halfedgesand insert the second diagram instead — in compliance with the rotation.The fact that this operation is well-defined, is proved in the same way as forthe vertex multiplication in Γ. Because antisymmetry is presupposed we do

6.5. Vogel’s algebra Λ 147

not need to take the alternating average over the six ways of insertion, likein Γ, — all the six summands will be equal to each other.

Example.

1 2

3

∨1 2

3

=

1 2

3

.

Conjecturally, the antisymmetry requirement in this definition is super-fluous:

6.5.3. Conjecture. Λ = X 3, i.e. any fixed diagram with 3 legs is antisym-metric with respect to leg permutations.

Multiplication in the algebra Λ naturally generalizes to the action of Λon different spaces generated by 1- and 3-valent diagrams, e. g. the space ofopen diagrams B and the space of 3-graphs Γ. The same argument as aboveleads to the following theorem.

6.5.4. Theorem. The action of the algebra Λ on any space spanned bydiagrams modulo AS and IHX relations is well-defined.

6.5.5. Relation between Λ and Γ. Recall that by Γ≥1 we denoted thedirect sum of all homogeneous components of Γ except for Γ0:

Γ≥1 =∞⊕

n=1

Γn.

The vector space Γ≥1 is an algebra with respect to vertex multiplication.

There are two naturally defined maps between Λ and Γ≥1:

• Λ→ Γ≥1. Add three half-edges and a vertex to an element of Λ in agreementwith the leg numbering:

1

2

3

7→ .

• Γ≥1 → Λ. Choose an arbitrary vertex of a 3-graph, delete it together with thethree adjacent half-edges and antisymmetrize:

7→ 1

6

[ 1

23

−2

13

−1

32

−3

21

+

2

31

+

3

12].


It is fairly evident that these maps are mutually inverse (in particular,this implies that the map Γ≥1 → Λ is well-defined). We thus arrive at thefollowing result.

Proposition. The vector spaces Γ≥1 and Λ are isomorphic.

It is evident by definition that under this isomorphism the multiplicationin Λ corresponds to the vertex multiplication in Γ, therefore we have:

Corollary. The algebra Λ is isomorphic to (Γ≥1,∨).

Remark. If conjecture 6.5.3 is true for k = 3 then all the six terms(together with their signs) in the definition of the map Γ≥1 → Λ are equalto each other. This means that there is no need to antisymmetrize. Whatwe do is remove one vertex (with a small neighbourhood) and number thethree legs obtained according to their cyclic ordering at the deleted vertex.Also this would simplify the definition of the vertex multiplication in section6.3 because in this case

G = G ,

and we truly insert one graph in a vertex of another.

6.5.6. Some conjectures. Conjecture. ([Vo]) The algebra Λ is gener-ated by the elements t and xk with odd k = 3, 5, ...:

1 2

3

1 2

3

1 2

3

1 2

3

· · ·

t x3 x4 x5 · · ·

Conjecture. ([Kn2]) The linear operator Λ → Λ of multiplication bythe element t has no kernel (cf. similar conjecture for the algebra of 3-graphs,p. 140).

If these two conjectures are true, then the structure of the algebra Λis fully known: J. Kneissler’s paper [Kn2] contains a complete list of allrelations in terms of the generators t and xk. (The relations are rathersophisticated.)

Exercises 149

Exercises

(1) Find explicitly a chain of IHX and AS relations that proves the followingequality in the algebra Γ of 3-graphs:

=

(2) ∗ Is it true that the operator of edge multiplication by the ‘bubble’ β

7→

(in the algebras C, B and Γ) has no kernel (this is the Conjecture 6.2.10)?

(3) Let τ2 : X 2 ∈ X 2 denote the transposition of legs in a fixed diagram.Prove that τ2 is an identity. Hint: (1) prove that a hole can be draggedthrough a trivalent vertex (as in Lemma 6.2.3, where x is empty), (2)to change the numbering of the two legs, use manoeuvres like in Lemma6.2.4 with y = ∅).

(4) ∗ Let Γ be the algebra of 3-graphs.• Is it true that Γ has no divisors of 0 (this is the Conjecture 6.2.11)?• Is it true that Γ is generated by plane graphs?• Find generators and relations of the algebra Γ.• Suppose that a graph G ∈ Γ consists of two parts G1 and G2

connected by three edges. Is the following equality:

G1 G2 = G1 G2

(5) Let X k be the space of 1- and 3-valent graphs with k numbered legs.Consider a transposition of two legs of an element of X k.• Give a example of a non-zero element of X k with even k which

changes under such a transposition.• ∗ Is it true that any such transposition changes the sign of the ele-

ment if k is odd? (This is the Conjecture 6.5.3. The first nontrivialcase is when k = 3.)

(6) ∗ Let Λ be Vogel’s algebra, i.e. the subspace of X 3 consisting of allantisymmetric elements.• Is it true that Λ = X 3 (this is the case k = 3 of the Conjecture

6.5.3)?• Is it true that Λ is generated by the elements t and xk (this is the

Conjecture 6.5.6; see also Exercises 7 and 8)?


• Is it true that the operator of multiplication by t has no kernel (thisis the Conjecture 6.5.6)?

(7) Let t, x3, x4, x5, ... be the elements of the space X 3 defined above.• Prove that xi’s belong to Vogel’s algebra Λ, i.e. that they are

antisymmetric with respect to permutations of legs.• Prove the relation x4 = −4

3 t ∨ x3 − 13 t∨4.

• Prove that xk with an arbitrary even k can be expressed throught, x3, x5, ...

(8) Prove that the dodecahedron

d =

1 2

3

belongs to Λ, and express it as a vertex polynomial in t, x3, x5, x7, x9.

(9) ∗ Let X 3 be the space of 1- and 3-valent graphs with 3 numbered vertices.The group S3 acts in X 3, splitting it into 3 subspaces:• symmetric, which is isomorphic to B3 (open diagrams with 3 legs),• totally antisymmetric, which is Vogel’s Λ by definition, and• some subspace Q, corresponding to a 2-dimensional irreducible re-

presentation of S3.Question: is it true that Q = 0?

Chapter 7

Lie algebra weight

systems

Given a Lie algebra g equipped a non-degenerate invariant bilinear form,one can construct a weight system with values in the center of the universalenveloping algebra U(g). One can likewise define a similar operator fromthe space B into the ad-invariant part of the symmetric algebra S(g). Theseconstructions are due to M. Kontsevich [Kon1], with basic ideas alreadyappearing in [Pen]. If additionally we have a finite dimensional representa-tion of the Lie algebra then taking the trace of the corresponding operatorwe get a numeric weight system. It turns out that these weight systems arethe symbols of the quantum group invariants (Sec. 3.7.6). The constructionof weight systems based on representations first appeared in D. Bar-Natan’spaper [BN0].

7.1. Lie algebra weight systems for the algebra AGiven a metrized Lie algebra g, one can construct a weight system ϕg withvalues in the universal enveloping algebra U(g) (Kontsevich’s construction[Kon1]), whereas every linear representation V provides a weight systemwith values in the ground field (Bar-Natan’s construction [BN0, BN1]).The reader is invited to consult the Appendix for basic information on Liealgebras and universal envelopes.

7.1.1. Universal Lie algebra weight systems. Kontsevich’s construc-tion proceeds as follows. Let g be a metrized Lie algebra (see A.1.1. Choosea basis e1, . . . , em of the g and denote by e∗i the dual basis with respectto the metric (the ad-invariant non-degenerate bilinear form 〈·, ·〉). Given a

151

152 7. Lie algebra weight systems

chord diagram D with n chords, assign to each chord an independent integervariable (supposed to range from 1 to m, the dimension of the Lie algebra).Choose a base point * on the circle of the diagram D, different from the

chord endpoints, and, starting from this point and going in the positive di-rection, write out an indexed monomial by the following rule: meeting anendpoint of a chord for the first time, write ei, meeting second endpointof the same chord, write e∗i , where i is the index assigned to that chord.Then take the sum of mn terms like this, making every index run from 1 tom independently. Denote by ϕg(D) the resulting element of the universalenveloping algebra U(g).

For example,

ϕg

( )=

m∑

i=1

eie∗i =: c

is the quadratic Casimir element associated to the chosen invariant form.

Another example: if

D =*

ij

k

,

then

ϕg(D) =m∑

i=1

m∑

j=1

m∑

k=1

eiejeke∗i e∗ke∗j .

7.1.2. Theorem. The above construction has the following properties:

(1) the element ϕg(D) does not depend on the choice of the base pointon the diagram,

(2) it does not depend on the choice of the basis ei of the Lie algebra,

(3) it belongs to the ad-invariant subspace U(g)g of the universal en-veloping algebra U(g) (which coincides with the center ZU(g)).

(4) the function D 7→ ϕg(D) satisfies 4-term relations,

(5) the resulting mapping ϕg : A → ZU(g) is a homomorphism ofalgebras (but not of coalgebras).

Proof. (1) This property is closely related, and proved similarly to the factthat A, the algebra of chord diagrams on the circle, is isomorphic to thealgebra of chord diagrams on the line Al (see section 4.8) and the isomor-phism does not depend on the place of the cut. The proof uses property(4), which makes sense, because the diagrams that appear in a 4T relation,admit a coherent labeling of their chords (although there is no such coherentlabeling for all diagrams at once).

7.1. Lie algebra weight systems for the algebra A 153

(2) An exercise in linear algebra: take two different bases ei and fjof g and reduce the expression for ϕg(D) in one basis to the expression inanother using the transition matrix between the two bases.

(3) It is enough to prove that ϕg(D) commutes with any basic elementer. By property (2), we can choose the basis to be orthonormal with respectto the ad-invariant form 〈·, ·〉, so that e∗i = ei for all i. Now, the commutatorof er and ϕg(D) can be expanded into the sum of 2n expressions, similar toϕg(D), only with one of the ei’s replaced by the corresponding commutatorwith er. Due to antisymmetry of the structure constants cijk (Lemma A.1.6),these expressions cancel in pairs that correspond to the ends of each chord.

To take a concrete example,

[er,∑

ij

eiejeiej ]

=∑

ij

[er, ei]ejeiej +∑

ij

ei[er, ej ]eiej +∑

ij

eiej [er, ei]ej +∑

ij

eiejei[er, ej ]

=∑

ijk

crikekejeiej +∑

ijk

crjkeiekeiej +∑

ijk

crikeiejekej +∑

ijk

crjkeiejeiek

=∑

ijk

crikekejeiej +∑

ijk

crjkeiekeiej +∑

ijk

crkiekejeiej +∑

ijk

crkjeiekeiej .

Here the first and the second sums cancel with the third and the fourthsums, respectively.

(4) Any of the 3 pairwise differences of chord diagrams that constitutethe 4 term relation as in equation (3) (page 83) goes into

∑cijk . . . ei . . . ej . . . ek . . .

where cijk are structure constants of the algebra g.

(5) Using property (1), we can place the base point in the productdiagram D1 · D2 between D1 and D2. Then the identity ϕg(D1 · D2) =ϕg(D1)ϕg(D2) becomes evident.

The center ZU(g) of the universal enveloping algebra can also be denotedby Ug(g) (meaning the g-invariant part of U(g)), because an element of U(g)that commutes with all elements of g belongs to the center.

Remark. If D is a chord diagram with n chords, then

ϕg(D) = cn + terms of degree less than 2n in U(g).Indeed, we can permute the endpoints of chords on the circle without chang-ing the highest term of ϕg(D) because all additional summands arising ascommutators have degrees less than 2n. Hence the highest term of ϕg(D)


does not depend on D. Finally, if D is a diagram with n isolated chords,i.e. the n-th power of the diagram with one chord, then ϕg(D) = cn.

According to the Harish-Chandra theorem (see [Hum]) for a semi-simpleLie algebra g, the center ZU(g) is isomorphic to the algebra of polynomialsin certain variables c1 = c, c2, . . . , cr, where r = rank(g). The degreesd1 = 2, d2, . . . , dr of c = c1, c2, . . . , cr with respect to the natural filtrationof U(g) are well-defined.

7.1.3. sl2-weight system. Consider the Lie algebra sl2 of 2 × 2 matriceswith zero trace. It is a three dimensional Lie algebra spanned (as a vectorspace) by the matrices

H =

(1 00 −1

), E =

(0 10 0

), F =

(0 01 0

)

with the commutators

[H,E] = 2E, [H,F ] = −2F, [E,F ] = H .

We will use the symmetric bilinear form 〈x, y〉 = Tr(xy):

〈H,H〉 = 2, 〈H,E〉 = 0, 〈H,F 〉 = 0, 〈E,E〉 = 0, 〈E,F 〉 = 1, 〈F, F 〉 = 0.

One can easily check that it is ad-invariant. The corresponding dual basisis

H∗ =1

2H, E∗ = F, F ∗ = E.

So the Casimir element c = 12HH + EF + FE.

The center ZU(sl2) is isomorphic to the algebra of polynomials in asingle variable c. So ϕsl2(D) is a polynomial in c. In this section, following[CV], we explain a combinatorial procedure to compute this polynomial fora given chord diagram D.

The algebra sl2 is simple, therefore any invariant form is λ〈·, ·〉. Thecorresponding Casimir element cλ, as an element of the universal envelopingalgebra, is related to the old one c by the formula cλ = c

λ . Therefore, theold weight system

ϕsl2(D) = cn + an−1cn−1 + an−2c

n−2 + · · ·+ a2c2 + a1c

and the new one

ϕsl2,λ(D) = cnλ + an−1,λcn−1λ + an−2,λc

n−2λ + · · ·+ a2,λc

2λ + a1,λcλ

are related by the formula ϕsl2,λ(D) = 1λn · ϕsl2(D)|

c=λ·cλ, or

an−1 = λan−1,λ, an−2 = λ2an−2,λ, . . . a2 = λn−2a2,λ, a1 = λn−1a1,λ.


Theorem. Let ϕsl2(·) be the weight system associated to sl2 and theinvariant form 〈·, ·〉, take a chord diagram D and choose a chord a of D.Then

ϕsl2(D) = (c− 2k)ϕsl2(Da) + 2∑

1≤i<j≤k

(ϕsl2(Di,j)− ϕsl2(D

×i,j)),

where: k is the number of chords which intersect the chord a;Da is the chord diagram obtained from D by deleting the chord a;Di,j and D×i,j are the chord diagrams which are obtained from Da inthe following way.

Consider an arbitrary pair of chords ai and aj which are different from thechord a and such that each of them intersects a. The chord a divides thecircle into two arcs. Denote by ei and ej the endpoints of ai and aj whichbelong to the left arc and by e∗i , e

∗j the endpoints of ai and aj which belong to

the right arc. There are three ways to connect four points ei, e∗i , ej , e

∗j by two

chords. In Da we have the case (ei, e∗i ), (ej , e

∗j ). Let Di,j be the diagram with

the connection (ei, ej), (e∗i , e∗j ). Let D×i,j be the diagram with the connection

(ei, e∗j ), (e

∗i , ej). All other chords are the same in all diagrams:

D =ei

ej

e∗i

e∗j

ai

aj

a ; Da =ei

ej

e∗i

e∗j; Di,j =

ei

ej

e∗i

e∗j; D×i,j =

ei

ej

e∗i

e∗j.

The theorem allows one to compute ϕsl2(D) recursively because each ofthe three diagrams Da, Di,j and D×i,j has one chord less than D.

Examples. 1. ϕsl2

( )= (c−2)c. In this case, k = 1 and the sum

in the theorem is zero, since there are no pairs (i, j).

2. ϕsl2

( )= (c− 4)ϕsl2

( )+ 2ϕsl2

( )− 2ϕsl2

( )

= (c− 4)c2 + 2c2 − 2(c− 2)c = (c− 2)2c.

3. ϕsl2

( )= (c− 4)ϕsl2

( )+ 2ϕsl2

( )− 2ϕsl2

( )

= (c− 4)(c− 2)c+ 2c2 − 2c2 = (c− 4)(c− 2)c.


Remark. Choosing another invariant form λ〈·, ·〉, we obtained a modi-fied relation

ϕsl2,λ(D) =(cλ −

2k

λ

)ϕsl2,λ(Da) +

2

λ

∑

1≤i<j≤k

(ϕsl2,λ(Di,j)− ϕsl2,λ(D

×i,j)).

If k = 1, then the second summand vanishes. In particular, for the Killingform (λ = 4) and k = 1 we have

ϕg(D) = (c− 1/2)ϕg(Da).

It is interesting that the last formula is valid for any simple Lie algebra g

with the Killing form and any chord a having only one intersection withother chords. See exercise 8 for a generalization of his fact.

Exercise. Deduce the theorem form the following lemma by induction(in case of difficulty see the proof in [CV]).

Lemma. (6-term relations for the universal sl2 weight system) Letϕsl2(·) be the weight system associated to sl2 and the invariant form 〈·, ·〉.Then

ϕsl2

(− − +

)= 2ϕsl2

(−

);

ϕsl2

(− − +

)= 2ϕsl2

(−

);

ϕsl2

(− − +

)= 2ϕsl2

(−

);

ϕsl2

(− − +

)= 2ϕsl2

(−

).

These relations also prove a recursive way to compute ϕsl2(D) because thetwo chord diagrams of the right-hand side have one chord less than thediagrams of the left-hand side, and the last three diagrams of the left-handside are simpler than the first one since they have less intersections betweentheir chords. See Sec. 7.2.2 for a proof of this Lemma.

7.1.4. Weight system associated with a representation. Bar-Natan’sconstruction is obtained from Kontsevich’s one as follows.

A linear representation is a homomorphism of Lie algebras T : g →End(V ). It extends to a homomorphism of associative algebras U(T ) :U(g)→ End(V ). The composition of three maps (the last one is trace)

A ϕg→ U(g)U(T )→ End(V )

Tr→ C


by definition gives the weight system associated with the representation

ϕVg = Tr U(T ) ϕg.

If the representation T is irreducible, then, according to the Schur Lemma[Hum], every element of the center ZU(g) is represented by a scalar opera-tor λ · idV . Therefore its trace equals ϕTg (D) = λ dimV . The weight system

represented by the eigenvalue λ =ϕT

g (D)

dimV is obviously multiplicative withrespect to the product of chord diagrams.

Example: algebra sl2 with the standard representation. Con-sider the standard 2-dimensional representation St of sl2. Then the Casimirelement is represented by a matrix

c =1

2HH + EF + FE =

(3/2 00 3/2

)=

3

2· id2

In degree 3 we have the following weight systems

D

ϕsl2(D) c3 c3 c2(c− 2) c(c− 2)2 c(c− 2)(c− 4)

ϕStsl2(D) 27/4 27/4 −9/4 3/4 15/4

ϕ′Stsl2(D) 0 0 0 12 24

Here the last row represents an unframed weight system obtained formϕStsl2

(·) by the deframing procedure from Sec. 4.6.3. A comparison of this

computation with the one from Sec. 3.7.2 shows that symb(j3) = −12ϕ′Stsl2

.See exercises 14 and 15 at the end of the chapter for more information aboutthese weight systems.

7.1.5. glN with standard representation. Consider the Lie algebra g =glN of allN×N matrices and its standard representation St. Fix the trace ofthe product of matrices as the preferred ad-invariant form: 〈x, y〉 = Tr(xy).

The algebra glN is generated by matrices eij with 1 on the intersection

of i-th row with j-th column and zero elsewhere. We have 〈eij , ekl〉 = δliδkj ,

where δ is the Kronecker delta. Therefore, duality between glN and (glN )∗

defined by 〈·, ·〉 is given by the formula e∗ij = eji.

One can check that [eij , ekl] 6= 0 only in the following cases:

• [eij , ejk] = eik, if i 6= k,

• [eij , eki] = −ekj , if j 6= k,

• [eij , eji] = eii − ejj , if i 6= j,


This gives the following formula for the Lie bracket as a tensor from gl∗N ⊗gl∗N ⊗ glN :

[·, ·] =N∑

i,j,k=1

(e∗ij ⊗ e∗jk ⊗ eik − e∗ij ⊗ e∗ki ⊗ ekj).

D. Bar-Natan found the following elegant way how to compute the weightsystem ϕStglN

(·).

Theorem [BN0]. Denote by s(D) the number of connected componentsof the curve obtained by doubling all chords of a chord diagram D.

.

Then ϕStglN(D) = N s(D) .

Example. For D = we obtain the . So s(D) = 2, and

ϕStglN(D) = N2.

Proof. For each chord we attach a matrix eij to one endpoint and the matrixeji to another endpoint. The pair (ij) is an index associated with the givenchord; different chords have different indices that vary in an independentway. We can label the two copies of the chord as well as the four pieces ofthe Wilson loop adjacent to its endpoints, by indices i and j as follows:

eij ejii

j

i

j

i

j.

To compute the value of the weight system ϕStglN(D), we must sum up the

products . . . eijekl . . . . Since we deal with the standard representation ofglN , the product should be understood as genuine matrix multiplication,unlike the case of weight systems with values in the universal envelopingalgebra. Since eij · ekl = δjk · eil, we obtain a non-zero summand only ifj = k. This means that the labels of the chords must follow the pattern:

eij ejli j j l

i j j l

.

Therefore, all the labels on one and the same connected component of thecurve obtained are equal. Now, if we take the whole product of matricesalong the circle, we will get the operator eii whose trace is 1. Then wesum up all these ones over all possible labelings. The number of labelingsis equal to the number of possibilities to assign an index i, j, l,. . . to each


connected component of the curve obtained by doubling all the chords. Eachcomponent giving exactly N possibilities, the total number becomes N s(D).

Proposition. The weight system ϕStglN(D) depends only on the inter-

section graph of D.

Proof. Induction on the number of chords in D. If all chords are isolated,then the assertion is trivial. Otherwise, pick up two intersecting chords αand β in D and double them according to the theorem. Straightening thepinced circle with two double chords into a new Wilson loop, we get a chorddiagramD′′ having two chords less thanD. One can see that the intersectiongraph Γ(D′′) depends only on the intersection graph Γ(D).

α

β=

Indeed, for any two vertices i and j different from α, β their connectivity(by an edge) in Γ(D′′) either coincides with their connectivity in Γ(D) orchanges to the opposite depending on their location with respect to α andβ according to the rules:

location of i and j in Γ(D) connectivity of i and j in Γ(D′′)i and j are connected with both α and β the same

one of i and j is connected with both α and β,and another one is connected with only one ofthem

opposite

one of i and j is connected with both α and β,and another one is not connected with them

the same

both i and j are connected with one and the

same of α and βthe same

one of i and j is connected with one of α andβ, and another one is connected with anotherone

opposite

one of i and j is connected with one of α andβ, and another one is not connected with them

the same

none of i and j is connected with α and β the same

See exercise 9 at the end of this chapter for another proof of this propo-sition.

7.1.6. slN with standard representation. Here we describe a weightsystem ϕStslN

(D) associated with the Lie algebra slN , the invariant form

〈x, y〉 = Tr(xy), and its standard representation by N × N matrices withzero trace.


Following Sec. 3.7.2, introduce a state s for a chord diagram D as anarbitrary function on the set [D] of chords of D with values in the set1,− 1

N . With each state s we associate an immersed plane curve obtainedfrom D by resolutions of all its chords according to s:

c , if s(c) = 1; c , if s(c) = − 1

N.

Let |s| denote the number of components of the curve obtained in this way.

Theorem. ϕStslN(D) =

∑

s

(∏

c

s(c))N |s| , where the product is taken

over all n chords of D, and the sum is taken over all 2n states for D.

One can prove this theorem in the same way as we did for glN pickingup the appropriate basis for the vector space slN and then working withthe product of matrices (see exercise 13). However, we prefer to prove itin a different way, via several reformulations using the algebra structure ofweight systems which is dual to the coalgebra structure of chord diagrams(Sec. 4.6)

Reformulation 1. For a subset J ⊆ [D] (the empty set and the whole[D] are allowed) of chords of D, let DJ be the chord diagram formed bychords from J , let s(DJ) denote the number of connected components of thecurve obtained by doubling of all chords of DJ , |J | be the number of chordsin J , and n−|J | = |J | stands for the number of chords in J = [D]\J . Then

ϕStslN(D) =

∑

J⊆[D]

(−1)n−|J |N s(DJ )−n+|J |.

This assertion is obviously equivalent to the theorem where, for everystate s the subset J consisting of all chords c with value s(c) = 1.

Consider the weight system e−11N from Sec. 4.6.3, which is equal to the

constant 1(−N)n on any chord diagram with n chords.

Reformulation 2.

ϕStslN= e−

11N · ϕStglN

.

Indeed, by the definition of the product of weight systems (Sec. 4.6),(e−

11N · ϕStglN

)(D) :=

(e−

11N ⊗ ϕStglN

)(δ(D)) ,


where δ(D) is the coproduct (Sec. 4.5) of the chord diagram D. It splitsD into two complementary parts DJ and DJ : δ(D) =

∑J⊆[D]

DJ ⊗ DJ .

The weight system ϕStglN(DJ) gives N s(DJ ). The remaining part is given

by e−11N (DJ).

Reformulation 3.

ϕStglN= e

11N · ϕStslN

.

The equivalence of this formula to the previous one follows from the fact

that the weight systems e−11N and e

11N are inverse to each other as elements

of the completed algebra of weight systems.

Proof. We will prove the theorem in reformulation 3. The Lie algebra glNis a direct sum of slN and the trivial one-dimensional Lie algebra generatedby the identity matrix idN . Its dual is id∗N = 1

N idN . So we can choose a basisfor the vector space glN consisting of the basis for slN and the unit matrixidN . To every chord we must assign either a pair of dual basic elementsof slN , or the pair (idN ,

1N idN ), which is equivalent to forgetting the chord

and multiplying the obtained diagram by 1N . This precisely means that we

are applying the weight system e11N to the chord subdiagram DJ formed by

forgotten chords, and the weight system ϕStslNto the chord subdiagram DJ

formed by the remaining chords.

7.1.7. soN with standard representation. In this case a state s for D isa function on the set [D] of chords of D with values in the set 1/2,−1/2.The rule of resolution of a chord according to its state is

c , if s(c) =1

2; c , if s(c) = −1

2.

As usual, |s| denotes the number of components of the obtained curve.

Theorem [BN0, BN1]. ϕStsoN(D) =

∑

s

(∏

c

s(c))N |s| , where the

product is taken over all n chords of D, and the sum is taken over all 2n

states for D.

We leave the proof of this theorem to the reader as an exercise (number16 at the end of the chapter).


Here is the table of values of ϕStsoN(D) for basic diagrams of small degree:

D

ϕsoN (D) 12(N2−N) 1

4(N2−N) 1

8(N2−N) 1

8N(−N2+4N−3)

D

ϕsoN (D) 116N(3N2−8N+5) 1

8N(N−1)2 1

16N(N3−4N2+6N−3)

Exercises 17—21 contain additional information about this weight sys-tem.

7.1.8. sp2N with standard representation. It turns out that ϕStsp2N(D) =

(−1)n+1ϕso−2N (D), where the last notation means the formal substitution of−2N instead of the variable N in the polynomial ϕsoN (D), and n, as usual,means the number of chords of D. This implies that the weight systemϕStsp2N

does not provide any new knot invariant. Some details about it can

be found in [BN0, BN1].

It would be interesting to find a combinatorial description of weightsystems for the exceptional Lie algebras E6, E7, E8, F4, G2.

7.2. Lie algebra weight systems for the algebra CWe would like to extend the weight system ϕg(·) to the weight system ρg(·)for the algebra C in such a way that the diagram

Aϕg //

λ

U(g)

Cρg

77ppppppppppppp

becomes commutative. Here λ is the isomorphism of A and C from section5.3.1. (Since A and C are isomorphic, the word ‘extension’ is not quiteappropriate; we use it because the set of closed diagrams generating C isstrictly wider than the set of chord diagrams generating A and we are goingto explain the rule of computing the value of ρ on any closed diagram.)

The STU relation (Sec. 5.1.2), defining the algebra C, gives us a hint.Namely, if we assign elements ei, ej to the endpoints of chords of the T- and

7.2. Lie algebra weight systems for the algebra C 163

U- diagrams from the STU relations,

ei ej

e∗i e∗j

T−

ej ei

e∗i e∗j

U=

[ei,ej ]

e∗i e∗j

S,

then it is natural to assign the commutator [ei, ej ] to the trivalent vertex inthe S-diagram. The formal construction goes as follows.

For every closed Jacobi diagram C ∈ CDn let V = v1, . . . , vm be theset of its external vertices (lying on the Wilson loop) ordered according tothe orientation of the loop. We construct a tensor Tg(C) ∈ g⊗m whose i-th tensor factor g corresponds to the element vi of the set V . Then we putρg(C) to be equal to the image of the tensor Tg(C) in U(g) under the naturalprojection. This is the weight system for the algebra C.

To construct the tensor Tg(C) we cut all the edges between the trivalentvertices of C. This splits C into a union of elementary pieces (tripods), eachconsisting of one trivalent vertex and three univalent vertices. Here is anexample:

With each elementary piece we associate a copy of the tensor −J ∈ g⊗g⊗g

whose factors correspond to the univalent vertices in agreement with thecyclic ordering of the edges. This tensor is defined as follows. Consider theLie bracket [·, ·] as an element of g∗ ⊗ g∗ ⊗ g. Identification of g∗ and g via〈·, ·〉 provides the tensor J ∈ g⊗g⊗g. Due to the properties of [·, ·] and 〈·, ·〉this tensor J is skew-symmetric under the permutations of the three tensorfactors which ensures that the previous construction makes sense. Althoughit does not really matter, please note that we associate the tensor −J , notJ , to each vertex. This convention proves useful in the case g = glN : itmakes the graphical algorithm for the calculation of the weight system lookmore natural.

The tensor Tg(C) corresponding to C is combined from these elementaryones in the following way. Let us restore C from the tripod pieces. Eachtime two univalent vertices are put together, we contract the two tensorfactors corresponding to these vertices by taking the form 〈·, ·〉 on them.For example, if −J =

∑αj ⊗βj ⊗γj , then the tensor we relate to the union

of two pieces will be:

γj αk

αj

βj

γk

βk

∑

i,j,k

〈γj , αk〉αj ⊗ βj ⊗ βk ⊗ γk.


Gluing and contracting in such a way over all the edges previously cutwe obtain an element of the tensor product of several copies of g whosefactors correspond to the univalent vertices of C. This is what we denoteby Tg(C).

Speaking pictorially, we suspend a copy of the tensor cube of g with thedistinguished element −J living inside it at every internal vertex of C, thentake the tensor product over all internal vertices and compute the completecontraction of the resulting tensor, applying the bilinear form 〈·, ·〉 to everypair of factors that correspond to the endpoints of one internal edge, forexample:

1

2

4

3

g

BBBB

BBBB

BBBB

BBBB

g

||||

||||

||||

||||

g

gg⊗

g

gg⊗

g

||||||||

||||||||g

BBBBBBBB

BBBBBBBB

−→ g⊗ g⊗ g⊗ g ,

where the numbers at univalent vertices on the graph correspond to theorder of tensor factors from left to right in the result space g⊗4. Doing thiswith all internal vertices and edges we get the tensor Tg(C) ∈ g⊗m

Note that the correspondence C → Tg(C) satisfies the AS and IHXrelations:

(1) the AS relation follows from the fact that the tensor J changes signunder odd permutations of the three factors in g⊗ g⊗ g.

(2) the IHX relation is a corollary of the Jacobi identity in g.

Moreover, the element ρg(C), which is the image of the tensor Tg(C) underthe natural projection g⊗m → U(g), satisfies also the STU relation:

(3) the STU relation follows from the definition of commutator in theuniversal enveloping algebra.

To exemplify this construction we prove the following lemma relatingthe tensor corresponding to a ‘bubble’ with the quadratic Casimir tensor.

7.2.1. Lemma. For the Killing form 〈·, ·〉K as the preferred invariant formwe have

Tg( ) = Tg( ) .


For the bilinear form µ〈·, ·〉K the rule changes as follows: Tg( ) =

1µTg( ).

Proof. The cut and glue procedure above gives the following tensor writtenin an orthonormal basis ei.

ei

ek ej′

ej ek′

ei′∑

i,i′

∑

k,j,k′,j′

cijkci′j′k′〈ek, ej′〉K〈ej , ek′〉Kei ⊗ ei′

=∑

i,i′

(∑

j,k

cijkci′kj

)ei ⊗ ei′ ,

where cijk are the structure constants: J =d∑

i,j,k=1

cijkei ⊗ ej ⊗ ek.

To compute the coefficient(∑j,k

cijkci′kj

)let us find the value of the

Killing form 〈ei′ , ei〉K = Tr(adei′adei). Since adei(ej) =

∑k

cijkek and

adei′(ek) =

∑l

ci′klel, the (j, l)-entry of the matrix of the product adei′adei

will be∑k

cijkci′kl. Therefore 〈ei′ , ei〉K =∑j,k

cijkci′kj . Orthonormality of the

basis ei implies that∑j,k

cijkci′kl = δi,i′ . This means that the tensor in the

right-hand side equals∑iei⊗ ei, which is the quadratic Casimir tensor from

the left-hand side.

7.2.2. sl2-weight system. The main theorem about the sl2 weight systemon the algebra C is the following

Theorem ([CV]). For the invariant form 〈x, y〉 = Tr(xy) the followingequality of tensors holds.

Tsl2

( )= 2Tsl2

( )− 2Tsl2

( ).

If the chosen invariant form is λ〈·, ·〉, then the coefficient 2 in this equa-tion is replaced by 2

λ .


Proof. For the algebra sl2 the Casimir tensor and the Lie bracket tensorare

C =1

2H ⊗H + E ⊗ F + F ⊗ E ;

−J = −H⊗F⊗E+F⊗H⊗E+H⊗E⊗F−E⊗H⊗F−F⊗E⊗H+E⊗F⊗H.

Then the tensor corresponding to the right-hand side is (we numerate thevertices according to tensor factors)

Tsl2

( 1

2

4

3

)= −H⊗F⊗H⊗E+H⊗F⊗E⊗H+F⊗H⊗H⊗E

−F⊗H⊗E⊗H−H⊗E⊗H⊗F+H⊗E⊗F⊗H+E⊗H⊗H⊗F−E⊗H⊗F⊗H+2F⊗E⊗F⊗E−2F⊗E⊗E⊗F−2E⊗F⊗F⊗E+2E⊗F⊗E⊗F

= 2(

14H⊗H⊗H⊗H+ 1

2H⊗E⊗F⊗H+ 1

2H⊗F⊗E⊗H+ 1

2E⊗H⊗H⊗F

+E⊗E⊗F⊗F+E⊗F⊗E⊗F+ 12F⊗H⊗H⊗E+F⊗E⊗F⊗E+F⊗F⊗E⊗E

)

−2(

14H⊗H⊗H⊗H+ 1

2H⊗E⊗H⊗F+ 1

2H⊗F⊗H⊗E+ 1

2E⊗H⊗F⊗H

+E⊗E⊗F⊗F+E⊗F⊗F⊗E+ 12F⊗H⊗E⊗H+F⊗E⊗E⊗F+F⊗F⊗E⊗E

)

= 2Tsl2

( )− 2Tsl2

( ).

Remarks. 1. During the reduction of a closed diagram accordingto this theorem a closed circle different from the Wilson loop may occur (seethe example below). In this situation we replace the circle by the numericmultiplier 3 = dim sl2 which is the trace of the identity operator in theadjoint representation of sl2.

2. In the context of weight systems this relation was first noted in [CV].Then it was rediscovered several times. But in more general context ofgraphical notation for tensors it was known to R. Penrose [Pen] circa 1955.In a certain sense, this relation goes back to Euler and Lagrange becauseit is essentially equivalent to the classical “bac − cab” rule, a × (b × c) =b(a · c)− c(a · b), for the ordinary cross product of vectors in 3-space.


Example.

ρsl2

( )= 2ρsl2

( )− 2ρsl2

( )= 4ρsl2

( )

−4ρsl2

( )− 4ρsl2

( )+ 4ρsl2

( )

= 12c2 − 4c2 − 4c2 + 4c2 = 8c2 .

The next corollary implies the 6-term relation from Sec. 7.2.2.

Corollary.

ρsl2

( )= 2ρsl2

(−

); ρsl2

( )= 2ρsl2

(−

);

ρsl2

( )= 2ρsl2

(−

); ρsl2

( )= 2ρsl2

(−

).

7.2.3. glN with standard representation. For a closed diagram C ∈ Cnwith the set IV of t internal trivalent vertices we double each internal edge(not a piece of the Wilson loop) and count the number of components of theobtained curve as before. The only problem here is how to connect the linesnear an internal vertex. This can be decided by means of a state functions : IV → −1, 1.

Theorem [BN1]. Let ρStglN(·) be the weight system associated to the Lie

algebra glN , its standard representation, and the invariant form 〈x, y〉 =Tr(xy).

For a closed diagram C and a state s : IV → −1, 1 double everyinternal edge and connect the lines together in a neighborhood of a vertexv ∈ IV according to the state s:

v , if s(v) = 1; v , if s(v) = −1.

Let |s| denote the number of components of the curve obtained in this way.Then

ρStglN(C) =

∑

s

(∏

v

s(v))N |s| ,

where the product is taken over all t internal vertices of C, and the sum istaken over all 2t states for C.

Proof. A straightforward way to prove this theorem is to use the STUrelation and the theorem of Sec. 7.1.5. However we prefer a different waybased on the graphical notation for tensors developed below.


Using the identification of (glN )∗ with glN and generators eij indicatedat the beginning of Sec. 7.1.5 we can rewrite the Lie bracket as

J =N∑

i,j,k=1

(eji ⊗ ekj ⊗ eik − eji ⊗ eik ⊗ ekj).

To each internal trivalent vertex we associate the tensor

−J =N∑

i,j,k=1

(eji ⊗ eik ⊗ ekj − eji ⊗ ekj ⊗ eik) .

Graphical notation for tensors. Here we introduce a system ofgraphical notation for a special class of the elements of (glN )⊗n similar tothat invented by R. Penrose in [Pen] (see also [BN1]). Consider a plane dia-gram (T -diagram) that consists of n pairs of mutually close points arrangedalong a circle and connected pairwise by n lines. Strictly speaking, a T -diagram is a set X of cardinality 2n endowed with two involutions withoutfixed points (or, in other words, two partitions of X into n two-point sub-sets). In the pictures below, X is the set of all protruding endpoints of thelines which indicate the second partition, while the pairs of the first partitionare the endpoints that are drawn close to each other.

Given a T -diagram, we can write out the corresponding element of then-th tensor power of glN in the following way. We put the same index ateither end of each line, and then, traveling around the encompassing circle,write a factor eij each time we encounter a pair of neighboring points, thefirst of which is assigned the index i and the second, the index j. Then, weconsider the sum of all such tensors when all the indices range from 1 to N .In a sense, this procedure gives a graphical substitute for the formal Einsteinsummation rule in multi-index expressions. We must admit that in generalthe element of ⊗nglN thus obtained may depend on the choice of the startingpoint of the circle, — but the corresponding element of the symmetric power,obtained by symmetrization, will not depend on this choice.

For example, to find the tensor corresponding to the diagram we

take three indices i, j and k and write:ji

i k

jk

N∑

i,j,k=1

eji ⊗ eik ⊗ ekj .

Here is another example:ji

k j

ki

N∑

i,j,k=1

eji ⊗ ekj ⊗ eik.


These two examples are sufficient to graphically express the tensor J :

J = −

With every trivalent vertex of an open diagram we associate a copy ofthe tensor −J : −

depending on the value of state on the vertex. After all such resolutionswe get a linear combination of 2t tensor monomials representing the tensorTglN

(C). Then taking the corresponding element in the universal envelop-ing algebra U(glN ) and the trace of its standard representation we get thepolynomial weight system ρStglN

(C). Now repeating the same arguments as

at the end of the proof on p.158 we get the Theorem.

Example. Let us compute the value ρStglN

( ). There are four

resolutions of the triple points:Qs(v)=1

|s|=4

Qs(v)=−1

|s|=2

Qs(v)=−1

|s|=2

Qs(v)=1

|s|=2

Therefore, ρStglN

( )= N4 −N2.

Other properties of the weight system ρStglN(·) are formulated in exercises

28 - 32.

7.2.4. soN with standard representation. Now a state for C ∈ Cn willbe a function s : IE → −1, 1 on the set IE of internal edges (those whichare not on the Wilson loop). The value of a state indicates the way ofdoubling the corresponding edge:

e, if s(e) = 1;

e, if s(e) = −1.

In a neighborhood of a trivalent vertex we connect the lines in a standardfashion. For example, if the values of the state on three edges e1, e2, e3meeting at a vertex v are s(e1) = −1, s(e2) = 1, and s(e3) = −1, then wehave

ve1e2 e3

.

As usual, |s| denotes the number of components of the curve obtained inthis way.


Theorem [BN1]. Let ρStsoN(·) be the weight system associated to the Lie

algebra soN , its standard representation, and the invariant form 〈x, y〉 =Tr(xy). Then

ρStsoN(C) = 2#(IV )−#(IE)

∑

s

(∏

e

s(e))N |s| ,

where the product is taken over all internal edges of C, the sum is taken overall states for C, and #(IV ), #(IE) denote the numbers of internal verticesand edges respectively.

The Theorem can be reformulated as follows. For an internal vertex vconsider one of the following 8 resolutions (two for each edge incident to v)together with the corresponding sign:

+ − − −

+ + + −

Now let s be the choice of one these resolution for every internal triva-lent vertex, and sign(s) be the product of the corresponding signs over allinternal trivalent vertices. Then

ρStsoN(C) =

1

4#(IV )

∑

s

sign(s)N |s| ,

In this form the Theorem can be easily proved by using STU relationand the theorem of Sec. 7.1.7.

Example.

ρStsoN

( )= 1

4(N3 −N2 −N2 −N2 +N +N +N −N)

= 14(N3 − 3N2 + 2N) = 1

4N(N − 1)(N − 2) .

7.3. Lie algebra weight systems for the algebra BIn this section for a metrized Lie algebra g we construct a weight systemψ : B → S(g), defined on the space of open diagrams B and taking values inthe completes symmetric algebra of the vector space g (in fact, even in itsg-invariant subspace Sg(g)).

Let O ∈ OD be an open diagram. Denote by V the set of its legs(univalent vertices). Just as in Sec. 7.2 we construct a tensor

Tg(O) ∈⊗

v∈Vg,

7.3. Lie algebra weight systems for the algebra B 171

where the symbol⊗v∈V

g means the tensor product whose factors correspond

to the elements of the set V : it is defined like the usual tensor product,only instead of linearly ordered arrays (g1, ..., gm), gi ∈ g we use families ofelements of g indexed by the set V .

A numbering ν of the set of univalent vertices V yields an isomorphism

between⊗v∈V

g and the conventional tensor powerm⊗i=1

g = g⊗m (with a linear

order of the factors). This isomorphism takes Tg(O) into T νg (O). The averageover all numberings ν is the element of the symmetric power that we wantedto construct:

ψg(O) =1

m!

∑

ν∈Sm

T νg (O) ∈ Sm(g).

Algorithmically, this step means that we forget the symbols of the tensorproduct and view the result as an ordinary commutative polynomial in thevariables that correspond to a basis of g.

7.3.1. The formal PBW theorem. The relation between the Lie algebraweight system for chord diagrams and for open diagrams is expressed by thefollowing theorem.

Theorem. For any metrized Lie algebra g the diagram

B ψg−−−−→ S(g)

χ

yyβg

A −−−−→ϕg

U(g)

is commutative.

Proof. The assertion becomes evident as soon as one recalls the definitionsof all the constituents of the diagram: the isomorphism χ between the alge-bras A and B described in section 5.7, the weight systems ϕg and ψg, definedin sections 7.1 and 7.3, and βg — the Poincare–Birkhoff–Witt isomorphismtaking an element x1x2...xn into the arithmetic mean of xi1xi2 ...xin over allpermutations (i1, i2, ..., in). Its restriction to the invariant subspace Sg(g) isan isomorphism with the center of U(g).

7.3.2. Example. Let g be the Lie algebra so3. It has a basis a, b, cwhich is orthonormal with respect to the Killing form 〈·, ·〉K and with thecommutators [a, b] = c, [b, c] = a, [c, a] = b. So as a metrized Lie algebraso3 is isomorphic to the euclidean 3-space with the cross product as a Lie


bracket. The tensor that we put in every 3-valent vertex in this case is

J = a ∧ b ∧ c= a⊗ b⊗ c+ b⊗ c⊗ a+ c⊗ a⊗ b− b⊗ a⊗ c− c⊗ b⊗ a− a⊗ c⊗ b.

Since the basis is orthonormal, the only way to get a non-zero elementin the process of contraction along the edges is to choose the same basicelement on either end of the edge. On the other hand, the formula forJ shows that in every vertex we must choose a summand with differentbasic elements corresponding to the 3 edges. This leads to the followingalgorithm to compute the tensor Tso3(O) for a given diagram O: one mustlist all 3-colorings of the edges of the graph by 3 colors a, b, c such that the3 colors in every vertex are always different, then sum the tensor productsof the elements written on the legs, each taken with the sign (−1)s, wheres is the number of negative vertices (i. e. vertices where the colors, readcounterclockwise, come in the negative order a, c, b).

For example, consider the diagram (the Pont-Neuf diagram with para-meters (1, 3) in the terminology of O. Dasbach [Da3], see also p.329 below):

O =

It has 18 edge 3-colorings, which can be obtained from the following threeby permutations of (a, b, c):

c*

ab

ca

bc

a

b

a

cc

cb

c*

ab

ca

bc

a

b

cb

ca

a

c*

ab

ca

bc

a

b

bc

ba c

In these pictures, negative vertices are marked by small empty circles. Writ-ing the tensors in the counterclockwise order, starting from the marked

7.3. Lie algebra weight systems for the algebra B 173

point, we get:

2(a⊗ a⊗ a⊗ a+ b⊗ b⊗ b⊗ b+ c⊗ c⊗ c⊗ c)+a⊗ b⊗ b⊗ a+ a⊗ c⊗ c⊗ a+ b⊗ a⊗ a⊗ b+b⊗ c⊗ c⊗ b+ c⊗ a⊗ a⊗ c+ c⊗ b⊗ b⊗ c+a⊗ a⊗ b⊗ b+ a⊗ a⊗ c⊗ c+ b⊗ b⊗ a⊗ a+b⊗ b⊗ c⊗ c+ c⊗ c⊗ a⊗ a+ c⊗ c⊗ b⊗ b.

This tensor is not symmetric with respect to permutations of the 4 tensorfactors (although — we note this in passing — it is symmetric with respectto the permutations of the 3 letters a, b, c). Symmetrizing, we get:

ψso3(O) = 2(a2 + b2 + c2)2.

This example shows that the weight system defined by the Lie algebraso3, is closely related to the 4-color theorem, see [BN3] for details.

7.3.3. The glN weight system for the algebra B. We are going toadapt the graphical notation for tensors from Sec. 7.2.3 to this situation.Namely, with every trivalent vertex of an open diagram O we associate tworesolutions (T -diagrams):

−

After all such resolutions we get a linear combination of 2t T -diagrams,where t is the number of trivalent vertices of O. Each diagram consists of mpairs of points and 2m lines connecting them, and represents a tensor fromglN⊗m. Permuting the tensor factors in glN

⊗m corresponds to interchangingthe pairs of points in the diagram:

Therefore, if we consider a T -diagram up to an arbitrary permutation ofthe pairs, it correctly defines an element of Un(glN )/Un−1(glN ) ≃ Sn(glN ).On a purely combinatorial level, a T -diagram considered up to an arbitrarypermutation of pairs of legs is a set of cardinality 2n with a splitting inton ordered pairs and a splitting into n unordered pairs. Thus, the linearcombination of T -diagrams which is considered up to such permutationsrepresents the element ψglN

(C).

The center of U(glN ). It is known [Zh] that ZU(glN ) is generated byN variables xj , 1 ≤ j ≤ N (generalized Casimir elements):

cj =N∑

i1,...,ij=1

ei1i2ei2i3ei3i4 . . . eij−1ijeiji1


as a free commutative polynomial algebra.

In the graphical notation

cj =

︸︷︷︸j pairs

.

In particular, c1 = =N∑i=1

eii is the unit matrix (note that it is not the unit

of the algebra S(g)), c2 = =N∑

i1,i2=1ei1i2ei2i1 is the quadratic Casimir

element.

It is convenient to extend the list c1, . . . , cN of our variables by settingc0 = N . The graphical notation for c0 will be a circle. (Indeed, a circle hasno legs, so it must correspond to an element of the 0-th symmetric powerof g, which is the ground field. When the index written on the circle runsfrom 1 to N , we must take the sum of ones in the quantity N , thus gettingN .)

Example.

ψglN

( )= − − +

= − − + = 2(c0c2 − c21).

7.4. Lie algebra weight systems for the algebra Γ

The construction of the Lie algebra weight system ηg(·) for the algebra of3-graphs proceeds in the same way as for the algebra C (Sec. 7.2), using thestructure tensor J . Since 3-graphs have no univalent vertices, this weightsystem takes values in the ground field C. For a graph G ∈ Γ we putηg(G) := Tg(G) ∈ g0 ∼= C.

When computing the weight systems ηslN (·), and ηsoN (·) it is importantto remember that their value on the circle (i.e. the unit of algebra Γ, orin other words, the 3-graph without vertices) is equal to the dimension ofthe corresponding Lie algebra, N2 − 1, and 1

2N(N − 1). However, when weapply state sum formulas form sections 7.2.3 and 7.2.4 for graphs of degree≥ 1 we replace every circle in the curve obtained by the resolutions just byN .

7.4.1. Changing the bilinear form. Tracing the construction of ηg(·), itis easy to see that the function ηg,λ(·), corresponding to the form λ〈·, ·〉 is

7.4. Lie algebra weight systems for the algebra Γ 175

proportional to ηg(·):ηg,λ(G) = λ−nηg(G)

for G ∈ Γn.

7.4.2. Proposition. Multiplicativity with respect to the edge prod-uct in Γ. For a simple Lie algebra g and any choice of the ad-invariantnon-degenerate symmetric bilinear form 〈·, ·〉 the function 1

dim gηg : Γ → C

is multiplicative with respect to the edge product in Γ.

Proof. This fact follows from the property that up to proportionality thequadratic Casimir tensor of a simple Lie algebra is the only ad-invariant,symmetric, non-degenerate tensor from g⊗ g.

Cut an arbitrary edge of the graph G1 and consider the tensor thatcorresponds to the obtained graph with two univalent vertices. This tensoris proportional to the quadratic Casimir tensor c ∈ g⊗ g:

a · c = a

dim g∑

i=1

ei ⊗ ei .

Now, ηg,K(G1) can be obtained by contraction these two tensor factors. Thisgives ηg,K(G1) = adim g because of orthonormality of the basis ei. So we

can find the coefficient a = 1dim g

ηg(G1). Similarly, for the second graph G2

we get the tensor 1dim g

ηg(G2)·c. Now, if we put together one pair of univalent

vertices of the graphs G1 and G2 thus cut, then the partial contraction of theelement c⊗c ∈ g⊗g⊗g⊗g will give an element 1

(dim g)2ηg(G1)·ηg(G2)c ∈ g⊗g.

But, on the other hand, this tensor equals 1dim g

ηg(G1 · G2)c ∈ g ⊗ g. This

proves the multiplicativity of the function 1dim g

ηg.

7.4.3. Compatibility with the edge action of Γ on C. Recall the de-finition of the edge action of 3-graphs on closed diagrams (see Sec. 6.4.1).We choose an edge in G ∈ Γ and an internal edge in C ∈ C, and then takethe connected sum of G and C along the chosen edges. In fact, this actiondepends on the choice of the connected component of C \ Wilson loopcontaining the chosen edge. It is well defined only on the primitive subspaceP ⊂ C. In spite of this indeterminacy we have the following lemma.

Lemma. For any choice of the gluing edges ρg(G · C) =ηg(G)dim g

ρg(C).

Indeed, in order to compute ρg(C) we assemble the tensor Tg(C) fromelementary piece tensors gluing them along the edges by contraction withthe quadratic Casimir tensor c. By the previous argument, to compute thetensor Tg(G · C) one must use the tensor 1

dim gηg(G) · c instead of simply c

for the chosen edge. This gives the coefficient 1dim g

ηg(G) in the expression

for ρg(G · C) as compared with ρg(C).


One particular case of this action is especially interesting: when thegraph G varies, while C is fixed and equal to Θ, the chord diagram withonly one chord. In this case the action is an isomorphism of the vector spaceΓ with the subspace P2 of the primitive space P generated by connectedclosed diagrams with 2 legs (section 6.4.2).

Corollary. For the weight systems associated to a simple Lie algebra g

and the Killing form 〈·, ·〉K :

ηg,K(G) = ρadg,K(G ·Θ) ,

where ρadg,K is the weight system corresponding to the adjoint representationof g.

Proof. Indeed, according to the Lemma, for the universal enveloping alge-bra invariants we have

ρg,K(G ·Θ) =1

dim gηg(G)ρg,K(Θ) =

1

dim gηg(G)

dim g∑

i=1

eiei ,

where ei is a basis orthonormal with respect to the Killing form. Now tocompute ρadg,K(G · Θ) we take the trace of the product of operators in theadjoint representation:

ρadg,K(G ·Θ) =1

dim gηg(G)

dim g∑

i=1

Tr(adeiadei) = ηg(G)

by the definition of the Killing form.

7.4.4. Proposition. Multiplicativity with respect to the vertex prod-uct in Γ. Let w : Γ → C be an edge-multiplicative weight system, andw(β) 6= 0. Then 1

w(β)w : Γ → C is multiplicative with respect to the ver-

tex product. In particular, for a simple Lie algebra g, 1ηg(β)ηg(·) is vertex-

multiplicative.

Proof. According to 6.3.4 the edge product is related to the vertex productlike this: G1 ·G2 = β · (G1∨G2) because of edge-multiplicativity. Therefore,

w(G1) · w(G2) = w(β · (G1 ∨G2)) = w(β) · w(G1 ∨G2) .

This means that the weight system 1w(β)w : Γ → C is multiplicative with

respect to the vertex product.

Corollary. The weight systems 12N(N2−1)

ηslN ,2

N(N−1)(N−2)ηsoN : Γ →C associated with the ad-invariant form 〈x, y〉 = Tr(xy) are multiplicativewith respect to the vertex product in Γ.

This follows from the direct computation on a “bubble”:ηslN (β) = 2N(N2 − 1), and ηsoN (β) = 1

2N(N − 1)(N − 2).

7.4. Lie algebra weight systems for the algebra Γ 177

7.4.5. Compatibility with the vertex action of Γ on C. The vertexaction G∨C of a 3-graph G ∈ Γ on a closed digram C ∈ C (see Sec. 6.4.3) isdefined as the alternating sum of 6 ways to glue the graph G with the closeddiagram C along chosen internal vertices in C and G. Again this action iswell defined only on the primitive space P.

Lemma. Let g be a simple Lie algebra. Then for any choice of thegluing vertices in G and C:

ρg(G ∨ C) =ηg(G)

ηg(β)ρg(C) .

Proof. Using the edge action (section 7.4.3) and its relation to the vertexaction (section 6.3.4) we can write

ηg(G)

dim gρg(C) = ρg(G · C) = ρg(β · (G1 ∨ C)) =

ηg(β)

dim gρg(G1 ∨ C) ,

which is equivalent to what we need.

7.4.6. slN - and soN -polynomials. In this subsection we consider weightsystems ηslN (·) supplied with the bilinear form 〈x, y〉 = Tr(xy), and ηsoN (·)supplied with the bilinear form 〈x, y〉 = 1

2Tr(xy). In soN case this formis more convenient since it gives polynomials with integral coefficients. Inparticular, for such a form ηsoN (β) = N(N − 1)(N − 2), and in the statesum formula from the Theorem of Sec. 7.2.4 the coefficient in front of thesum will be 1.

The polynomial ηslN (G) (= ηglN(G)) is divisible by 2N(N2−1) (exercise

39) and the quotient is a multiplicative function with respect to the vertexproduct. We call the quotient the reduced sl-polynomial and denote it by

sl(G).

Dividing the so-polynomial ηsoN (G) by N(N − 1)(N − 2) (see exercise40), we obtain the reduced so-polynomial so(G), which is also multiplicativewith respect to the vertex product.

A. Kaishev [Kai] computed the values of sl-, and so-polynomials onthe generators of Γ of small degrees (for so-polynomial the substitution


M = N − 2 is used).

deg sl-polynomial so-polynomial

1 β 1 1

4 ω4 N3+12N M3−3M2+30M−24

6 ω6 N5+32N3+48N M5−5M4+80M3−184M2+408M−288

7 ω7 N6+64N4+64N2 M6−6M5+154M4−408M3+664M2−384

8 ω8 N7+128N5+128N3

+192N

M7−7M6+294M5−844M4+1608M3−2128M2

+4576M−3456

9 ω9 N8+256N6+256N4

+256N2

M8−8M7+564M6−1688M5+3552M4−5600M3

−5600M3+6336M2+6144M−9216

10 ω10 N9+512N7+512N5

+512N3+768N

M9−9M8+1092M7−3328M6+7440M5−13216M4

+18048M3−17920M2+55680M−47616

10 δ N9+11N7+114N5

−116N3

M9−9M8+44M7−94M6+627M5+519M4

−2474M3−10916M2+30072M−17760

11 ω11 N10+1024N8+1024N6

+1024N4+1024N2

M10−10M9+2134M8−6536M7+15120M6

−29120M5+45504M4−55040M3+48768M2

+145408M−165888

One may find a lot of recognizable patterns in this table. For example,wee see that

sl(ωn) = Nn−1 + 2n−1(Nn−3 + · · ·+N2), for odd n > 5;

sl(ωn) = Nn−1 + 2n−1(Nn−3 + · · ·+N3) + 2n−23N , for even n ≥ 4.

It would be interesting to know if these observations make part of somegeneral theorems.

Later we will need the values of the - and so-polynomials on the “Mer-

cedes element” τ = = 12β

2. They are sl(τ) = N , and so(τ) = M =

N − 2.

7.5. Weight systems not coming from Lie algebras

In this section we shall briefly describe an idea of P. Vogel [Vo] how toprove the existence of a weight system which do not come from semisimpleLie algebras. Instead of Vogel’s Λ we will use its isomorphic copy, the algebraΓ≥1 with vertex multiplication. It is enough to construct an element fromC with the following properties:

• it is non-zero;

• it is primitive;

7.5. Weight systems not coming from Lie algebras 179

• it is homogeneous (belongs to Pn for a certain n);

• all semisimple Lie algebra weight systems vanish on it.

The construction consists of two steps. At the first step we will find anon-zero element of Γ on which all Lie algebra weight systems vanish. Werepresent this element as a polynomial in the generators of small degree withrespect to the vertex product. At the second step we construct the requiredelement of the algebra of closed diagrams C.

Consider the following three elements

Γ7 ∋ Xsl := 3ω6 ∨ τ − 6ω4 ∨ τ∨3 − ω∨24 + 4τ∨6 ;

Γ10 ∋ Xso := −108ω10 + 3267ω8 ∨ τ∨2 − 1920ω6 ∨ ω4 ∨ τ−20913ω6 ∨ τ∨4 + 372ω∨3

4 + 8906ω∨24 ∨ τ∨3

+13748ω3 ∨ τ∨6 − 3352τ∨9 ;

Γ7 ∋ Xex := 45ω6 ∨ τ − 71ω4 ∨ τ∨3 − 18ω∨24 + 32τ∨6 .

Using the multiplicativity with respect to the vertex product and the tablesfrom the previous subsection one can check that

sl(Xsl) = 0, and so(Xso) = 0.

Vogel [Vo] shows that weight systems associated to the exceptional Lie al-gebras vanish on Xex. Therefore, any semisimple Lie algebra weight systemvanishes on the product

X := Xsl ∨Xso ∨Xex ∈ Γ22 .

On the other hand, according to Vogel [Vo], X is a non-zero element of Γ,because there is a Lie superalgebra D(2, 1, α) weight system which takes anon-zero value on X.

The study of the Lie superalgebra weight systems is beyond the scopeof this book, so we restrict ourselves with a reference to [Kac] for a gen-eral theory of Lie superalgebras and to [FKV, Lieb] as regards the weightsystems coming from Lie superalgebras.

Now we are going to construct the required primitive element of C. Con-sider the “Mercedes closed diagram” represented by the same graph

t1 = ,

where the circle is understood as the Wilson loop and there is only oneinternal trivalent vertex. According to Sec. 6.4.3 the algebra of 3-graphs Γacts on the primitive space P via vertex multiplication. By Lemma 6.4.4the map G 7→ G ∨ t1 is an injection Γn → Pn+1. Therefore, the element

X ∨ t1


is a non-zero primitive element of degree 23 in the algebra of closed diagrams.

Because of the Lemma 7.4.5 all simple (and therefore semisimple) Liealgebra weight systems vanish on X∨t1. Therefore any weight system whichis non-zero on X ∨ t1 may not come from semisimple Lie algebras. In fact(see [Vo]), even the Lie superalgebra weight system which is non-zero on Xvanishes on X ∨ t1. So even Lie superalgebra weight systems are not enoughto generate all weight systems.

Exercises

(1) Let (g1, 〈·, ·〉1) and (g2, 〈·, ·〉2) be two metrized Lie algebras. Then theirdirect sum g1⊕g2 is also a metrized Lie algebra with respect to the form〈·, ·〉1 ⊕ 〈·, ·〉2. Prove that ϕg1⊕g2 = ϕg1 · ϕg2 .

The general aim of exercises (2)-(8) is to compare the behavior of ϕsl2 (D) with that

of the chromatic polynomial of a graph. In these exercises we use the form 2〈·, ·〉 as the

invariant form.

(2) ∗ Prove that ϕsl2(D) depends only on the intersection graph Γ(D) of thechord diagram D (see Sec. 4.3).

(3) Prove that the polynomial ϕsl2(D) has alternating coefficients.

(4) Show that for any chord diagram D the polynomial ϕsl2(D) is divisibleby c.

(5) ∗ Prove that the sequence of coefficients of the polynomial ϕsl2(D) isunimodal (i.e. its absolute values form a sequence with only one maxi-mum).

(6) Let D be a chord diagram with n chords for which Γ(D) is a tree. Provethat ϕsl2(D) = c(c− 2)n−1.

(7) Prove that the highest three coefficients of the polynomial ϕsl2(D) are

ϕsl2(D) = cn − e · cn−1 + (e(e− 1)/2− t+ 2q) · cn−2 − . . . ,

where e is the number of intersections of chords of D; t and q are thenumbers of triangles and quadrangles of D respectively. A triangle is asubset of three chords of D with all pairwise intersections. A quadrangleof D is an unordered subset of four chords a1, a2, a3, a4 which form acircle of length four. This means that, after a suitable relabeling, a1

intersects a2 and a4, a2 intersects a3 and a1, a3 intersects a4 and a2,a4 intersects a1 and a3 and any other intersections are allowed. For

Exercises 181

example,

e( )

= 6, t( )

= 4, q( )

= 1 .

(8) (A. Vaintrob). Define a vertex multiplication of chord diagrams as fol-lows:

* ∨ * := = .

Of course it depends of the choice of vertices where multiplication isperformed. Prove that for any choice

ϕsl2(D1 ∨D2) =ϕsl2(D1) · ϕsl2(D2)

c.

(9) (S. Lando, B. Mellor [Mel2]). Let s(D) bethe number of connected components of thecurve obtained by doubling all chords of achord diagram D, and N be a formal variable. Consider the adjacencymatrix M of the intersection graph of D as a matrix over the field oftwo elements F2 = 0, 1. Prove that s(D)− 1 is equal to the corank ofM (over F2), and deduce from this that ϕStglN

(D) depends only on the

intersection graph Γ(D).Essentially the same weight system was independently rediscovered

by B. Bollobas and O. Riordan [BR2] who used it to produce a poly-nomial invariant of ribbon graphs generalizing the Tutte polynomial[BR3].

(10) (D. Bar-Natan, S. Garoufalidis [BNG])

a). Prove that ϕStglNsatisfies the 2-term re-

lations.b). Show that the vector space of chord

diagrams modulo 2-term relations isspanned by (m1,m2)-caravans of m1

“one-humped camels” and m2 ‘two-humped camels”

For a generalization of these facts to ϕStsoN

see [Mel2].

=

=

2-term relations

m1 m2

(m1,m2)-caravan

(11) (D. Bar-Natan, S. Garoufalidis [BNG]) Prove that the symbol of the co-efficient cn of the Conway polynomial, as a weight system symb(cn)(D),equals to the determinant (over F2) of the adjacency matrix of the in-tersection graph Γ(D).


(12) Let Dn be a chord diagram with n chordswhose intersection graph is a circle. Provethat ϕStglN

(Dn) = ϕStglN(Dn−2). Deduce from

here that ϕStglN(Dn) = N2 for odd n, and

ϕStglN(Dn) = N3 for even n.

Dn =

n chords

(13) Work out a proof of the theorem from Sec. 7.1.6 about slN weight systemwith standard representation, similar to the one given in Sec. 7.1.5. Usethe basis of the vector space slN consisting of the matrices eij for i 6= jand the matrices eii − ejj for i < j.

(14) Prove that ϕ′StslN≡ ϕ′StglN

.

Hint. ϕ′StslN= e−

N2−1N

11 · ϕStslN= e−N11 · ϕStglN

= ϕ′StglN.

(15) Compare the symbol of the coefficient jn of the Jones polynomial (sec-tion 3.7.2) with the weight system coming form sl2, and prove that

symb(jn) =(−1)n

2ϕ′Stsl2

.

Hint. Compare the formula for ϕ′Stsl2from the previous problem and

the formula for symb(jn) from Sec. 3.7.2, and prove that

(|s| − 1) ≡ #chords c such that s(c) = 1 mod 2 .

(16) Work out a proof of the theorem from Sec. 7.1.7 about soN weight systemin standard representation. Use the basis of soN formed by matriceseij − eji for i < j. (In case of difficulty consult [BN0, BN1].)

(17) Work out a proof, similar to the proof of the Proposition from Sec. 7.1.5,that ϕStsoN

(D) depends only on the intersection graph of D.

(18) (B. Mellor [Mel2]). For any subset J ⊆ [D], let MJ denotes markedadjacency matrix of the intersection graph of D over the filed F2 , thatis the adjacency matrix M only the diagonal elements corresponding toelements of J are switched to 1. Prove that

ϕStsoN(D) =

Nn+1

2n

∑

J⊆[D]

(−1)|J |N−rank(MJ ) ,

where the rank is computed as the rank of a matrix over F2. Thisgives another proof that ϕStsoN

(D) depends only on the intersection graphΓ(D).

(19) Show that N = 0 and N = 1 are roots of polynomial ϕStsoN(D) for any

chord diagram D.

(20) Let D be a chord diagram with n chords, such that the intersectiongraph Γ(D) is a tree. Show that ϕStsoN

(D) = 12nN(N − 1).

Exercises 183

(21) Let Dn be a chord diagram from problem 12. Prove thata). ϕStsoN

(Dn) = 12

(ϕStsoN

(Dn−2)− ϕStsoN(Dn−1)

);

b). ϕStsoN(Dn) = 1

(−2)nN(N − 1)(an−1N − an) , where the recurrent

sequence an is defined by a0 = 0, a1 = 1, an = an−1+2an−2.

(22) Compute ρsl2

( ), ρsl2

( ), and show that these two closed

diagrams are linearly independent.

(23) Let tn ∈ Cn+1 be a closed diagram with n legsas shown in the figure.Show that ρsl2(tn) = 2nc.

tn =

n legs

(24) Let wn ∈ Cn be a wheel with n spokes.Show thatρsl2(wn) = 2cρsl2(wn−2)+2ρsl2(wn−1)−2n−1c .

wn =

n spokes

(25) Let p ∈ Pkn be a primitive element of degree n > 1 with at most kexternal vertices. Show that ρsl2(p) is a polynomial in c of degree 6

[k/2], where [k/2] means the integral part of k/2, i.e. greatest integerless than or equal to k/2.

Hint. Use the theorem from 7.2.2 and the calculation of ρsl2(t3) fromexercise (23).

(26) Let ϕ′sl2 be the deframing of the weight system ϕsl2 according to the

procedure of Section 4.6.3. Show that for any element D ∈ An, ϕ′sl2(D)is a polynomial in c of degree 6 [n/2]

Hint. Use the previous exercise, exercise (7) of Chapter 4, and Sec-tion 5.5.2.

(27) Denote by Vk the k-dimensional irreducible representation of sl2 (see

Appendix A.1.1). Let ϕ′Vksl2

be the corresponding weight system. Show

that for any element D ∈ An of degree n, ϕ′Vksl2

(D)/k is a polynomial ink of degree at most n.

Hint. The quadratic Casimir number in this case is k2−12 .

(28) Let C ∈ Cn (n > 1) be a closed diagram (primitive element) whichremains connected after deleting the Wilson loop. Prove thatρStglN

(C) = ρStslN(C).

Hint. For the Lie algebra glN the tensor J ∈ gl⊗3N lies in the subspace

sl⊗3N .

(29) Consider a closed diagram C ∈ Cn and a glN -state s for it (see p. 167).Construct a surface Σs(C) by attaching a disk to the Wilson loop, re-placing each edge by a narrow band and gluing the bands together at


trivalent vertices — flatly, if s = 1, and with a twist, if s = −1. Here isan example:

C =+

−−

+=: Σs(C) .

a). Show that the surface Σs(C) is orientable.b). Compute the Euler characteristic of Σs(C) in terms of C, and show

that it depends only on the degree n of C.c). Prove that ρStglN

(C) is an odd polynomial for even n, and it is an

even polynomial for odd n.

(30) Show that N = 0, N = −1, and N = 1 are roots of the polynomialρStglN

(C) for any closed diagram C ∈ Cn (n > 1).

(31) Compute ρStglN(tn), where tn is the closed diagram from problem 23.

Answer. For n ≥ 1, ρStglN(tn) = Nn(N2 − 1).

(32) For the closed diagram wn as in the problem 24, prove that ρStglN(wn) =

N2(Nn−1 − 1) for odd n, and ρStglN(wn) = N(Nn +N2 − 2) for even n.

Hint. Prove the recurrent formula ρStglN(wn) = Nn−1(N2 − 1) +

ρStglN(wn−2) for n ≥ 3.

(33) Extend the definition of the weight system symb(cn)(·) of the coefficientcn of the Conway polynomial to Cn, and prove that

symb(cn)(C) =∑

s

(∏

v

s(v))δ1,|s| ,

where the states s are precisely the same as in the theorem of Sec. 7.2.3for the weight system ρStglN

(·). In other words, prove that symb(cn)(C)

equals to the coefficient at N in the polynomial ρStglN(C). In particular,

show that symb(cn)(wn) = −2 for even n, and symb(cn)(wn) = 0 forodd n.

(34) Show that N = 0, and N = 1 are roots of the polynomial ρStsoN(C) for

any closed diagram C ∈ Cn (n > 0).

(35) a). Let C ∈ C be a closed diagram with at least one internal trivalentvertex. Prove that N = 2 is a root of the polynomial ρStsoN

(C).

b). Deduce that ρStso2(C) = 0 for any primitive closed diagram C.

Hint. Consider the eight states that differ only on three edges meet-ing at an internal vertex (see p.170). Show that the sum over these eight

states,∑

sign(s)2|s|, equals zero.

(36) Prove that ρStsoN(tn) = N−2

2 ρStsoN(tn−1) for n > 1.

In particular, ρStsoN(tn) = (N−2)n

2n+1 N(N − 1).

Exercises 185

(37) Using some bases in A2 and B2, find the matrix of the isomorphism χ,then calculate (i.e. express as polynomials in the standard generators)the values on the basic elements of the weight systems ϕg and ψg for theLie algebras g = so3 and g = glN and check the validity of the relationβ ψ = ϕ χ in this particular case.

(38) Prove that the mapping ψ : B → S(g) is well-defined, i.e. that the IHXrelation follows from the Jacobi identity.

(39) Show that N = 0, N = −1, and N = 1 are roots of the polynomialηglN

(G) for any 3-graph G ∈ Γn (n > 1).

(40) Show that N = 0, N = 1 and N = 2 are roots of polynomial ηsoN (G)for any 3-graph G ∈ Γn (n > 0).

Part 3

The Kontsevich

Integral

Chapter 8

The Kontsevich

integral

The Kontsevich integral was invented by M. Kontsevich [Kon1] as a toolto prove the fundamental theorem of the theory of Vassiliev invariants(Th. 4.2.1). Any Vassiliev knot invariant can be factored through the uni-versal invariant defined by the Kontsevich integral.

Detailed (and different) expositions of the construction and properties ofthe Kontsevich integral can be found in [BN1, CD3, Les]. Other relevantpapers of primary importance are [Car1], [LM1], [LM2].

8.1. First examples

We start with two examples where the Kontsevich integral appears in asimple form with a clear geometrical meaning. In these examples, as well asin the general construction of the Kontsevich integral, we represent 3-spaceR3 as the product of a real line R with coordinate t and a complex plane Cwith complex coordinate z.

8.1.1. The braiding number of a 2-braid. A braid with two stringshas a complete integer invariant: the number of half-twists that one strandmakes around another.

z (t)1z (t)2

189

190 8. The Kontsevich integral

If the strands are parameterized by complex-valued functions z1(t) andz2(t) of a real variable t, 0 ≤ t ≤ 1, then this number can be computed bythe integral formula

1

2πi

∫ 1

0

dz1 − dz2z1 − z2

.

8.1.2. Kontsevich type formula for the linking number. The Gaussintegral formula for the linking number of two spatial curves lk(K,K ′)(Sec. 2.2.2) involves integration over the torus (the product of the two givencurves). In the formula we are going to give here the integration is spreadover an interval only. This formula generalizes the one for the braids quotedabove and represents one coefficient of the full Kontsevich integral in thissituation.

Theorem. Suppose that two given curves are embedded in R3 = Rt×Cz asMorse curves in the sense that the function t on K ∪K ′ has only nondegen-erate critical points and all its critical values are distinct.

z (t) jj z (t)’

Then

lk(K,K ′) =1

2πi

∫

m<t<M

∑

j

(−1)↓jd(zj(t)− z′j(t))zj(t)− z′j(t)

,

where m and M are the minimum and the maximum values of t on thelink K ∪ K ′, j is the index that enumerates all possible choices of a pairof strands on the link as functions zj(t), z

′j(t) corresponding to K and K ′,

respectively, and the integer ↓j is the number of chosen strands which areoriented downwards.

Proof. ...................

The Kontsevich integral can be regarded as a far-going generalization ofthis formula. Here, we kept track of one horizontal chord moving along thetwo curves. The full Kontsevich integral aims at encoding all informationabout how sets of horizontal chords on the knot (or a tangle) rotate whenmoved in the vertical direction. From a more general viewpoint, the Kont-sevich integral represents the monodromy of the Knizhnik–Zamolodchikov

8.2. The construction 191

connection in the complement to the union of diagonals in Cn (we will dis-cuss that in section 12.1).

8.2. The construction

Represent three-dimensional space R3 as the direct product of a complexline C with coordinate z and a real line R with coordinate t. The integralis defined for Morse knots, i. e. knots K embedded in R3 = Cz ×Rt in sucha way that the coordinate t is a Morse function on K (all critical pointsare nondegenerate and all critical levels are different). Its values belong

to the graded completion A′ of the algebra of chord diagrams with 1-termrelations A′ = A/(Θ). (By definition, the elements of a graded algebra arefinite linear combinations of homogeneous elements. The graded completionconsists of all infinite combinations of such elements.)

8.2.1. Definition. The Kontsevich integral Z(K) of the knot K is definedas follows:

Z(K) =∞∑

m=0

1

(2πi)m

∫

tmin<tm<···<t1<tmaxtj are noncritical

∑

P=(zj ,z′j)(−1)↓PDP

m∧

j=1

dzj − dz′jzj − z′j

.

The ingredients of this formula have the following meaning.

The real numbers tmin and tmax are the minimum and the maximum ofthe function t on K.

The integration domain is the m-dimensional simplex tmin < tm < · · · <t1 < tmax divided by the critical values into a certain number of connectedcomponents. For example, for the following embedding of the unknot andm = 2 the corresponding integration domain has six connected componentsand looks like

t

tmax

tc1tc2

tmin

z

t2

tmax

tc1

tc2

tmin

t1tmaxtc1tc2tmin

The number of summands in the integrand is constant in each connectedcomponent of the integration domain, but can be different for different com-ponents. In each plane t = tj ⊂ R3 choose an unordered pair of distinctpoints (zj , tj) and (z′j , tj) on K, so that zj(tj) and z′j(tj) are continuous func-

tions. We denote by P = (zj , z′j) the set of such pairs for j = 1, . . . ,m.


The integrand is the sum over all choices of the pairing P . In the exampleabove for the component tc1 < t1 < tmax, tmin < t2 < tc2 we have onlyone possible pair of points on the levels t = t1 and t = t2. Therefore,the sum over P for this component consists of only one summand. Unlikethis, in the component tc2 < t1 < tc1 , tmin < t2 < tc2 we still have onlyone possibility for the level t = t2, but the plane t = t1 intersects our

knot K in four points. So we have(42

)= 6 possible pairs (z1, z

′1) and the

total number of summands is six (see the picture below).

For a pairing P the symbol ‘↓P ’ denotes the number of points (zj , tj) or(z′j , tj) in P where the coordinate t decreases along the orientation of K.

Fix a pairing P . Consider the knot K as an oriented circle and connectthe points (zj , tj) and (z′j , tj) by a chord. We obtain a chord diagram with m

chords. The corresponding element of the algebra A′ is denoted by DP . Inthe picture below, for each connected component in our example, we showone of the possible pairings, the corresponding chord diagram with the sign(−1)↓P and the number of summands of the integrand (some of which areequal to zero in A′ due to the one-term relation).

1

6 summands

(−1)

1 summand

2 (−1) 2

2

21

36 summands

(−1)

1 summand

(−1)

6 summands

(−1)

1 summand

(−1)

Over each connected component, zj and z′j are smooth functions in tj .

By

m∧

j=1


we mean the pullback of this form to the integration do-

main of variables t1, . . . , tm. The integration domain is considered with theorientation of the space Rm defined by the natural order of the coordinatest1, ..., tm.

By convention, the term in the Kontsevich integral corresponding tom =0 is the (only) chord diagram of order 0 with coefficient one. It representsthe unit of the algebra A′.

8.3. Basic properties 193

8.3. Basic properties

The Kontsevich integral has the following basic properties (the proofs willbe given later):

• Z(K) converges for any Morse knot K.

• It is invariant under deformations of the knot in the class of Morseknots.

• It behaves in a simple way under the deformations that add a pairof new critical points to a Morse knot.

Let us explain the last item in more detail. Unfortunately, the Kontse-vich integral is not invariant under deformations that change the number ofcritical points of the function t, therefore it does not define a knot invariant.However, the following formula shows how the integral changes under suchdeformations:

(1) Z

( )= Z(H) · Z

( ).

Here the first and the third pictures depict two embeddings of an arbitraryknot, differing only in the shown fragment,

H =

is the hump (unknot embedded in R3 in the specified way), and the product

is the product in the completed algebra A′ of chord diagrams. The lastequality allows one to define a genuine knot invariant by the formula

I(K) =Z(K)

Z(H)c/2,

where c denotes the number of critical points of K and the ratio means the

division in the algebra A′ according to the rule (1+a)−1 = 1−a+a2−a3+. . .

The expression I(K) is sometimes referred to as the ‘final’ Kontsevichintegral as opposed to the ‘preliminary’ Kontsevich integral Z(K). It is in-variant under arbitrary deformations of K and in fact represents a universalVassiliev invariant in the following sense.

Consider a function w on the set of chord diagrams with m chords satis-fying one- and four-term relations (an unframed weight system). Applyingthis function to the m-homogeneous part of the series I(K), we get a nu-merical knot invariant w(I(K)). This invariant is a Vassiliev invariant oforder m and any Vassiliev invariant can be obtained in this way (a proofwill be given in Section 8.8).


The Kontsevich integral behaves in a nice way with respect to the naturaloperations on knots, such as mirror reflection, changing the orientation ofthe knot, mutation of knots. In a different normalization it is multiplicativeunder the connected sum of knots:

I ′(K1#K2) = I ′(K1)I′(K2),

where

I ′(K) = Z(H)I(K) =Z(K)

Z(H)c/2−1.

For any knot K the coefficients in the expansion of Z(K) over an arbitrarybasis consisting of chord diagrams are rational [Kon1, LM2].

8.4. Example of calculation

We will calculate the coefficient of the chord diagram in Z(H), whereH is the hump (plane curve with 4 critical points, see p. 193) directly fromthe definition of the Kontsevich integral. The following computation is validfor an arbitrary shape of the curve, provided that the length of the segmentsa1a2 and a3a4 (see picture below) decreases with t1, while that of the segmenta2a3 increases.

First of all, note that out of the total number of 51 pairings shown in

the picture on p. 192, the following 16 contribute to the coefficient of :

We are, therefore, interested only in the band between the critical valuesc1 and c2. Denote by a1, a2, a3, a4 (resp. b1, b2, b3, b4) the four points ofintersection of the knot with the level t = t1 (resp. t = t2):

a4a3a2a1

b2b1 b3 b4

c

1

t

c2

t

z

t

1

2

8.4. Example of calculation 195

The sixteen pairings shown in the picture above correspond to the differentialforms

(−1)j+k+l+md ln ajk ∧ d ln blm,

where ajk = ak − aj , blm = bm − bl, and the pairs (jk) and (lm) cantake 4 different values each: (jk) ∈ (13), (23), (14), (24) =: A, (lm) ∈(12), (13), (24), (34) =: B. The sign (−1)j+k+l+m is equal to (−1)↓P , be-cause in our case upward oriented strings have even numbers, while down-ward oriented strings have odd numbers.

The coefficient of is therefore equal to

1

(2πi)2

∫

∆

∑

(jk)∈A

∑

(lm)∈B(−1)j+k+l+md ln ajk ∧ d ln blm

=− 1

4π2

∫

∆

∑

(jk)∈A(−1)j+k+1d ln ajk ∧

∑

(lm)∈B(−1)l+m−1d ln blm

=− 1

4π2

∫

∆

d lna14a23

a13a24∧ d ln

b12b34b13b24

,

where the integration domain ∆ is the triangle described by the inequalitiesc2 < t1 < c1, c2 < t2 < t1. Assume the following notation:

u =a14a23

a13a24, v =

b12b34b13b24

.

It is easy to see that u is a increasing function of t1 ranging from 0 to 1,while v is an decreasing function of t2 ranging from 1 to 0. Therefore, themapping (t1, t2) 7→ (u, v) is a diffeomorphism with a negative Jacobian, andafter the change of variables the integral we are computing becomes

1

4π2

∫

∆′

d lnu ∧ d ln v

where ∆′ is the image of ∆. It is obvious that the boundary of ∆′ containsthe segments u = 1, 0 ≤ v ≤ 1 and v = 1, 0 ≤ u ≤ 1 that correspond tot1 = c1 and t2 = c2. What is not immediately evident is that the third sideof the triangle ∆ also goes into a straight line, namely, u+ v = 1. Indeed, ift1 = t2, then all b’s are equal to the corresponding a’s and the required factfollows from the identity a12a34 + a14a23 = a13a24.

1

2

c1

c2 c1

t

t2

1

1

v

u0

c


Therefore,

1

4π2

∫

∆′

d lnu ∧ d ln v =1

4π2

1∫

0

1∫

1−u

d ln v

du

u

= − 1

4π2

1∫

0

ln(1− u)duu.

Taylorising the logarithm we get

1

4π2

∞∑

k=1

1∫

0

uk

k

du

u=

1

4π2

∞∑

k=1

1

k2=

1

4π2ζ(2) =

1

24.

Two things are quite remarkable in this answer: (1) that it is expressedthrough a value of the zeta function, and (2) that the answer is rational. Infact, for any knot K the coefficient of any chord diagram in Z(K) is rationaland can be computed through the values of multivariate ζ-functions:

ζ(a1, . . . , an) =∑

0<k1<k2<···<kn

k−a11 . . . k−an

n .

We will speak about it in more detail in Section 12.2.

A complete formula for Z(H) was found in [BLT] (see Section 10.6).

8.5. Convergence of the integral

8.5.1. Proposition. For any Morse knot K, the Kontsevich integral Z(K)converges.

Proof. The integrand of the Kontsevich integral may have singularities nearthe boundaries of the connected components. This happens only in thefollowing two cases.

1) An isolated chord (zj , z′j) tends to zero:

zj z′j

In this case the corresponding chord diagram DP is equal to zero in A′ bythe one-term relation.

2) A chord (zj , z′j) tends to zero near a critical point but is separated

from that point by another, ‘long’ chord (zj−1, z′j−1):

tc

tj zj z′j

zj−1

zc z′cz′j−1

z′′j

8.6. Invariance of the integral 197

In this case the integral for (j − 1)-th chord can be estimated as follows:∣∣∣∣∣∣∣

tc∫

tj

dzj−1 − dz′j−1

zj−1 − z′j−1

∣∣∣∣∣∣∣≤ C

∣∣∣∣∣∣∣

tc∫

tj

d(zj−1 − z′j−1)

∣∣∣∣∣∣∣= C

∣∣(zc − zj)− (z′c − z′′j )∣∣ ≤ C ′|zj − z′j |

for some positive constants C and C ′. So the integral for the chord numberj − 1 is as small as zj − z′j and this smallness compensates the smallness ofthe denominator corresponding to the j-th chord.

This implies the convergence of the Kontsevich integral.

8.6. Invariance of the integral

8.6.1. Theorem. The Kontsevich integral is invariant under the deforma-tions in the class of knots all of whose critical points are nondegenerate (butcritical values are allowed to coincide).

A deformation is understood as a smooth family of diffeomorphisms ofR3 which starts with the identity.

The proof of this theorem spans the whole of this section. It is based ontwo propositions (8.6.9 and 8.6.10) that we prove below.

Any deformation that keeps a given Morse knot within the class of knotswith nondegenerate critical points and hence preserves the number of criticalpoints can be decomposed into horizontal deformations and movements ofcritical points.

A horizontal deformation is a deformation of R3 which preserves allhorizontal planes t = const and leaves all the critical points fixed. Theinvariance under horizontal deformations is the most essential point of thetheory. We prove it in the next subsection.

A movement of the critical point C is a deformation which is identicaleverywhere outside a small neighborhood of C and does not introduce newcritical points on the knot. In subsection 8.6.11 we consider invariance underthe movements of critical points.

As we mentioned before, the Kontsevich integral is not invariant underdeformations that change the number of critical points. Its behavior undersuch deformations will be discussed in section 8.7.

8.6.2. Tangled chord diagrams. To prove the horizontal invariance ofthe Kontsevich integral, we will use tangles from 1.5 and chord diagrams onthem. The point is that a Morse knot can be cut into several tangles by


any finite set of non-critical horizontal planes, and the knot is equal to theproduct of all such tangles.

Definition. A tangled chord diagram (T -chord diagram) is a tangle T sup-plied with a set of chords considered up to an orientation- and component-preserving diffeomorphism of the tangle.

For example, the following two diagrams are equivalent:

=

Note also that non-equivalent tangles may carry equivalent tangled chorddiagrams, e.g.

6= as tangles

but = as tangled chord diagrams.

The product of tangles induces a multiplication of tangled chord dia-grams.

8.6.3. Definition. Let T be a tangle. The space of T -chord diagrams A′Tis a vector space spanned by all chord diagrams on T modulo the set oftangled one- and four-term relations, defined below.

Tangled one-term relation: A tangled chord diagram with an isolatedchord is equal to zero in A′T :

= 0 or = 0 .

Tangled four-term relation: Denote by uij the following tangledchord diagram consisting of n parallel vertical strings and one chord betweenthe i-th and j-th strings:

uij = (−1)↓

i j

··· ··· ···,

where ↓ stands for the number of endpoints of the chords in which the ori-entation is directed downwards. Then the tangled four-term relation reads:

[uij + uik, ujk] = 0


or, in pictures:

(1) (−1)↓

ji k

− (−1)↓

ji k

+ (−1)↓

ji k

− (−1)↓

ji k

= 0 .

Remark. By a permutation of the indices i, j, k we obtain another formof the tangled four-term relation:

(2) (−1)↓

kj i

− (−1)↓

kj i

+ (−1)↓

j ik

− (−1)↓

kj i

= 0 .

8.6.4. Exercise. If a tangle is closed into a circle in agreement with ori-entations of the strings, the tangled four-term relation carries over into theordinary four-term relation of Section 4.1.

Here is an example:

− − + = 0 .

If (bottom T1) = (top T2) then the multiplication of tangled chord dia-grams descends to the multiplication of quotient spaces

A′T1⊗A′T2

→ A′T1·T2.

8.6.5. Now we can consider the Kontsevich integral for tangles, taking val-ues in the space of tangled chord diagrams. It is defined by the same formulaas the Kontsevich integral for knots (see Section 8.2), with tmin and tmax

corresponding to the bottom and the top of the tangle, respectively. Thisalso provides a construction of the Kontsevich integral for braids, as braidsare a particular case of tangles (1.5.4).

8.6.6. Proposition. The Kontsevich integral for tangles is multiplicative:

Z(T1) · Z(T2) = Z(T1 · T2)

whenever the product T1 · T2 is defined.

This important fact is a direct consequence of the Fubini theorem aboutmultiple integrals. In particular it implies the vertical multiplicativity ofthe Kontsevich integral for Morse knots: if one Morse knot is put on topof the other (with respect to the coordinate t in R3 = R × C), then theKontsevich integral of their connected sum is equal to the product (in thesense of algebra A′) of Kontsevich integrals of the summands.

8.6.7. Exercise. Compute the integrals

R := Z

( )and R−1 := Z

( )


(this can be done in a straightforward way, directly from the definition).

Answer:

R = · exp(

2

), R−1 = · exp

(2

),

where the first factors mean tangled chord diagrams without chords. Theyare different as tangles, but equal to each other as tangled chord diagrams.

8.6.8. Invariance under horizontal deformations. Let us decomposethe given knot into a product of tangles without critical points of the func-tion t and (very-very thin) tangles containing the critical levels. Due tomultiplicativity, it is enough to prove two facts:

1) the Kontsevich integral for a tangle without critical points is invari-ant under horizontal deformations that preserve the top and the bottompointwise,

2) the Kontsevich integral for a tangle with a unique critical point isinvariant under horizontal deformations that preserve the critical point.

These two facts are proved in the following two propositions.

8.6.9. Proposition. Let T0 be a tangle without critical points and Tλ, ahorizontal deformation of T0 to T1 (preserving the top and the bottom of thetangle). Then Z(T0) = Z(T1).

Proof. Denote by ω the integrand form in the m-th term of the Kontsevichintegral:

ω =∑

P=(zj ,z′j)(−1)↓DP

m∧

j=1


.

Here the functions zj , z′j depend not only on t1, ..., tm, but also on λ, and

all differentials are understood as complete differentials with respect to allthese variables. This means that the form ω is not exactly the form whichappears in the Kontsevich’s integral (it has some additional dλ’s), but thisdoes not change the integrals over the simplices

∆λ = tmin < tm < · · · < t1 < tmax × λ,because the value of λ on such a simplex is fixed.

We must prove that the integral of ω over ∆0 is equal to its integral over∆1.

Consider the product polytope

∆ = ∆0 × [0, 1] =

∆0 ∆1

.


By Stokes’ theorem we have

∫

∂∆

ω =

∫

∆

dω .

The form ω is exact: dω = 0. The boundary of the integration domainis ∂∆ = ∆0 −∆1 +

∑ faces. The theorem will follow from the fact thatω|face = 0. To show this consider two types of faces.

The first type corresponds to tm = tmin or t1 = tmax . In this situationdzj = dz′j = 0 for j = 1 or m, because zj and z′j do not depend on λ.

The second type is that where we have tk = tk+1 for some k. In thiscase we have to choose the k-th and (k + 1)-th chords on the same levelt = tk. In general, the endpoints of these chords may coincide and wedon’t get a chord diagram at all. Strictly speaking, we have to define theprolongation of ω and DP to such a face more carefully. It is natural toprolong DP to this face as a locally constant function. This means that forthe case in which some endpoints of k-th and (k+1)-th chords belong to thesame string (and therefore coincide) we place k-th chord a little higher than(k+ 1)-th chord, so that its endpoint differs from the endpoint of (k+ 1)-thchord. This trick yields a well-defined prolongation of DP and ω to the face,and we use it here.

All summands of ω are divided into three parts:

1) k-th and (k + 1)-th chords connect the same two strings;

2) k-th and (k+1)-th chords are chosen in such a way that their endpointsbelong to four different strings;

3) k-th and (k + 1)-th chords are chosen in such a way that there existexactly three different strings containing their endpoints.

Consider all these cases one by one.

1) We have zk = zk+1 and z′k = z′k+1 or vice versa. So d(zk − z′k) ∧d(zk+1 − z′k+1) = 0 and therefore the restriction of ω to the face is zero.

2) All choices of chords in this part of ω appear in mutually cancelingpairs. Fix four strings and number them by 1, 2, 3, 4. Suppose that for acertain choice of the pairing, the k-th chord connects the first two stringsand (k+1)-th chord connects the last two strings. Then there exists anotherchoice for which on the contrary the k-th chord connects the last two stringsand (k + 1)-th chord connects the first two strings. These two choices givetwo summands of ω differing by a sign:

· · · d(zk − z′k) ∧ d(zk+1 − z′k+1) · · ·+ · · · d(zk+1 − z′k+1) ∧ d(zk − z′k) · · · = 0.

3) This is the most difficult case. The endpoints of k-th and (k + 1)-thchords have exactly one string in common. Call the three relevant strings


1, 2, 3 and denote by ωij the 1-formdzi − dzjzi − zj

. Then ω is the product of a

certain (m− 2)-form and the sum of the following six 2-forms:

(−1)↓ ω12 ∧ ω23 + (−1)↓ ω12 ∧ ω13

+(−1)↓ ω13 ∧ ω12 + (−1)↓ ω13 ∧ ω23

+(−1)↓ ω23 ∧ ω12 + (−1)↓ ω23 ∧ ω13 .

Using the fact that ωij = ωji, we can rewrite this as follows:

((−1)↓ − (−1)↓

)ω12 ∧ ω23

+

((−1)↓ − (−1)↓

)ω23 ∧ ω31

+

((−1)↓ − (−1)↓

)ω31 ∧ ω12 .

The tangled four-term relations (1), (2) say that the expressions in paren-theses are one and the same element of A′T , hence the whole sum is equalto(

(−1)↓ − (−1)↓)

(ω12 ∧ ω23 + ω23 ∧ ω31 + ω31 ∧ ω12).

The 2-form that appears here is actually zero! This simple, but remarkablefact, known as Arnold’s identity (see [Ar1]) can be put under the followingform:

f + g + h = 0 =⇒ df

f∧ dgg

+dg

g∧ dhh

+dh

h∧ dff

= 0

(in our case f = z1 − z2, g = z2 − z3, h = z3 − z1) and verified by a directcomputation.

Proposition 8.6.9 is proved.

8.6.10. Proposition. Let T0 and T1 be two tangles, each with exactly onecritical point, differing from each other by a horizontal deformation pre-serving the top and the bottom of the tangles and the critical point. ThenZ(T0) = Z(T1).


Proof. Assume for simplicity that the values of t at the bottom, the criticalpoint and the top are −1, 0 and 1, respectively. There exists a horizontaldeformation of T0 into T1 via a family of tangles Tλ, 0 ≤ λ ≤ 1, such thatTλ coincides with T0 for |t| < (1− λ)2 and coincides with T1 for |t| > 1− λ.We require that all Tλ be smooth tangles with only one critical point whichdoes not depend on λ. By Proposition 8.6.9 we have that Z(Tλ) = Z(T0)for any λ between 0 and 1. But the limit of the tangle Tλ, as λ goes to 1,is T1. This proves Proposition 8.6.10 and the horizontal invariance of theKontsevich integral (Theorem 8.6.1).

8.6.11. Moving the critical points. Let T0 and T1 be two tangles whichare identical except a sharp tail of width ε, which may be twisted:

T1T0

DDt

ε

More exactly, we assume that (1) T1 is different from T0 only inside a regionD which is the union of disks Dt of diameter ε lying in horizontal planeswith fixed t ∈ [t1, t2], (2) each tangle T0 and T1 has exactly one critical pointin D, and (3) each tangle T0 and T1 intersects every disk Dt at most in twopoints. We call the passage from T0 to T1 a special movement of the criticalpoint.

8.6.12. Theorem. The Kontsevich integral does not change under a specialmovement of the critical point: Z(T0) = Z(T1).

Proof. The difference between Z(T0) and Z(T1) can come only from theterms with a chord ending on the tail.

If the highest of such chords connects the two sides of the tail, then thecorresponding tangled chord diagram is zero by a one-term relation.

So we can assume that the highest, k-th, chord is a ‘long’ chord, whichmeans that it connects the tail with another part of T1. Suppose the end-point of the chord belonging to the tail is (z′k, tk). Then there exists anotherchoice for k-th chord which is almost the same but ends at another point of


the tail (z′′k , tk) on the same horizontal level:

zk zkzk zk

The corresponding two terms appear in Z(T1) with the opposite signs dueto (−1)↓.

Let us estimate the difference of the integrals corresponding to such k-thchords:

∣∣∣∣∣∣∣

tc∫

tk+1

d(ln(z′k − zk)) −tc∫

tk+1

d(ln(z′′k − zk))

∣∣∣∣∣∣∣=

∣∣∣∣∣ ln

(z′′k+1 − zk+1

z′k+1 − zk+1

)∣∣∣∣∣

=

∣∣∣∣∣ ln

(1 +

z′′k+1 − z′k+1

z′k+1 − zk+1

)∣∣∣∣∣ ∼∣∣z′′k+1 − z′k+1

∣∣ ≤ ε

(here tc is the value of t at the uppermost point of the tail).

Now, if the next (k + 1)-th chord is also long, then, similarly, it can bepaired with another long one so that they give a contribution to the integralproportional to

∣∣z′′k+2 − z′k+2

∣∣ ≤ ε.In the case the (k + 1)-th chord is short (i.e. connects two points z′′k+1,

z′k+1 of the tail) we have the following estimate for the double integral cor-responding to k-th and (k + 1)-th chords:

∣∣∣∣∣∣∣

tc∫

tk+2

( tc∫

tk+1

d(ln(z′k − zk)) −tc∫

tk+1

d(ln(z′′k − zk)))dz′′k+1 − dz′k+1

z′′k+1 − z′k+1

∣∣∣∣∣∣∣

≤ const ·

∣∣∣∣∣∣∣

tc∫

tk+2

∣∣z′′k+1 − z′k+1

∣∣ dz′′k+1 − dz′k+1∣∣z′′k+1 − z′k+1

∣∣

∣∣∣∣∣∣∣

= const ·

∣∣∣∣∣∣∣

tc∫

tk+2

d(z′′k+1 − z′k+1)

∣∣∣∣∣∣∣∼∣∣z′′k+2 − z′k+2

∣∣ ≤ ε .

Continuing such considerations, we see that the difference between Z(T0)and Z(T1) is O(ε). Now by horizontal deformations we can make ε tend tozero. This proves the theorem and completes the proof of the Kontsevich

8.7. Horizontal multiplicativity 205

integral’s invariance in the class of knots with nondegenerate critical points.

8.7. Horizontal multiplicativity

Here we will prove formula 1 (page 193) which describes the behavior ofthe Kontsevich integral under the addition of a pair of critical points; usingthis formula, we will construct the universal Vassiliev invariant (aka finalKontsevich’s integral) and discuss its various properties.

If two Morse knots are added together vertically, like this:

then the Kontsevich integral of the connected sum is the product of theKontsevich integrals of the summands — simply by the Fubini theorem ofstandard calculus. We must, however, be ableto compute the Kontsevichintegral of a horizontal sum of Morse knots that looks like this:

In the argument below, we will use the relation between the Kontsevichintegral of knots, of tangles and of the complements to tangles.

Suppose we have a Morse knot K with a distinguished tangle T .

T

t

ttop

tbottom


Let ttop and tbottom be the maximal and minimal values of t on the tangle T .Consider the integral ZT (K) defined by the same formula as the Kontsevichintegral Z(K) but with a reduced number of summands in the integrand forsome connected components of the integration domain (see p.191). Supposethat C is a connected component whose projection on the coordinate axis tjis entirely contained in the segment [tbottom, ttop]. Then we use only thosepairings P = (zj , z′j) on the level tj for which either both points (zj , tj)

and (z′j , tj) belong to the tangle T or neither of them does. This means that

we omit the summands of Z(K) which contain a ‘long’ chord (connectingT with the remaining part of K). This is done for every component C andevery variable tj such that all values of tj on C are between the top and thebottom of the tangle.

8.7.1. Definition. The integral ZT (K) will be referred to as the Kontsevichintegral of the knot K reduced with respect to the tangle T .

The following lemma was first explicitely stated by T. Le [Le].

8.7.2. Lemma. Reduction with respect to any tangle does not change thevalue of the Kontsevich integral: ZT (K) = Z(K).

Warning. The Lemma does not say anything about the integrals over aparticular component of the integration domain. It only claims the coinci-dence of the total integrals.

To prove the Lemma, we need the following observation which followsfrom the same arguments that we used in the proof of the horizontal invari-ance of the Kontsevich’s integral in section 8.6.8:

Let (Ks, Ts) be a one-parameter deformation of the pair (K = K0, T =T0) of a knot K with a distinguished tangle T in the class of Morse knots.Then the reduced integral is invariant: ZTs(Ks) = ZT (K).

Proof of the lemma. We must prove that the difference Z(K)− ZT (K)is equal to zero.

Let us deform the knot K shrinking the tangle T sideways into a narrowrectangle box of width ε:

ε ≈l

Tε


We will prove that the difference Z(Kε)−ZTε(Kε) tends to zero as ε→ 0.Since both integrals are invariant under the deformation, this immediatelyimplies the lemma. Unfortunately this shrinking cannot be done by hori-zontal deformations alone because we have to drag critical points into thenarrow box. To perform the shrinking, we will first move all critical pointsof T into the box, using special movements (see p. 203) of critical points.Then, by a horizontal deformation, we shrink the whole tangle to the box.After that we return the critical points to their initial levels:

T

ε

εT

Applying a similar procedure to the portion of Kε \Tε that lies betweenthe horizontal planes t = tbottom and t = ttop, we can squeeze it into aparallelepiped of width ε, too. (Remark: The process under study involvesmoving the critical values, hence it makes no sense to consider the integralover one particular component. This is why the lemma is about the integralsover the whole integration domain.)

Now the difference Z(Kε) − ZTε(Kε) consists of summands of Z(Kε)with ‘long’ chords. So it is enough to prove that the integral correspondingto one ‘long’ chord tends to zero. For a long chord (zj , z

′j) connecting Tε

with the remaining part of Kε, the corresponding integral is equal to thelogarithm of the quotient of the differences zj − z′j at the upper and lower

limits of integration, which has the same order of magnitude as ln( ε+ll ) ∼ εand thus tends to zero when ε→ 0. The lemma is proved.

8.7.3. Example. Let us check directly that the lemma is true for the coef-ficient in the Kontsevich integral of the hump that we computed in section


8.4 using the following distinguished tangle:

We see that out of 16 summands that contribute to the coefficient of thecross chord diagram of order 2, we can leave only the following four:

To compute the integral for the sum of these four terms, we proceedexactly as we did in section 8.4. First, we introduce the notation:

b2

a2

c1

1tt2c2

t

z

a1 a3

b3b1

and put ajk = ak − aj , bjk = bk − bj . Then we write down the integraland find that — quite remarkably — the four summands also fold up neatlyinto one, and a similar change of variables takes it into exactly the samedilogarithm integral over the standard triangle that we obtained before:

1

(2πi)2

∫

∆

(−d ln a12 ∧ d ln b13 + d ln a12 ∧ d ln b23

+ d ln a13 ∧ d ln b13 − d ln a13 ∧ d ln b23)

=− 1

4π2

∫

∆

(−d ln

(a12

a13

)∧ d ln b13 + d ln

(a12

a13

)∧ d ln b23)

= − 1

4π2

∫

∆

d ln

(a12

a13

)∧ d ln

(b23b13

)=

1

4π2

∫

∆′

d lnu ∧ d ln v =1

24.

Here ∆ is the triangle c2 < t2 < t1 < c1, ∆′ is the triangle u ≤ 1,v ≤ 1, u + v ≥ 1, and the change of variables is given by the formulasu(t1, t2) = a12/a13, v(t1, t2) = b23/b13. The slanted side of the triangle ∆′

is indeed rectilinear and given by the equation u + v = 1 because, whent1 = t2, we have a12

a13+ b23

b13= a12

a13+ a23

a13= 1.


8.7.4. Theorem. We have

Z

( )= Z

( )· Z( )

,

where the first and the third pictures depict an arbitrary knot, differing onlyin the shown fragment, while the second picture represents an unknot embed-ded in R3 in the specified way and the product is the product in the completed

algebra A′ of chord diagrams.

To prove the theorem we will extend the construction of the Kontsevich

integral for a tangle Z(T ) ∈ A′T (see Section 8.6.5) to the complement ofthe tangle. For example, the complement K \T for the pair K, T shown onp. 205 looks like

top

bottom

We consider chord diagrams on K \ T made up of horizontal chordsmodulo one- and four-term relations which look exactly like tangled ones.

Now the Kontsevich integral Z(K \ T ) ∈ A′K\T can be defined in the sameway as for the tangles.

If T ′ is a tangle whose top and bottom coincide with the top and bottomof K \ T , then one can obviously define their product T ′ · (K \ T ) and theproduct of chord diagrams on them (concatenation product):

A′T ′ ×A′T\K → A′.

By definition, for any tangle T in the knot K

ZT (K) = Z(T ) · Z(K \ T ),

and lemma 8.7.2 implies

8.7.5. Lemma. (Tangled multiplication formula for the Konstevich inte-gral)

Z(K) = Z(T ) · Z(K \ T ).

We will investigate a little further the properties of the Kontsevich in-tegral for one-string tangles and their complements and then use them toprove the theorem.


A one-string tangle is a connected tangle that has only one string at thetop and at the bottom. For such tangles, as well as for their complements,we have two naturally defined closure operations:

• For the tangle.(1) Closure of the tangle T 7→ T . The tangle is converted into a

knot by adding an arc with two nondegenerate critical points.

(2) Closure of chord diagrams: the mapping A′T → A′defined by

the identification of the two endpoints of T . Let Z(T ) ∈ A′stand for the Kontsevich integral of the tangle closed in thisway.

• For the complement.(1) Closure of the complement K\T 7→ K \ T . The complement is

converted into a knot by adding an arc without critical points.

(2) Closure of chord diagrams: the mapping A′K\T → A′defined

by the identification of the two endpoints of K \ T .

Let Z(K \ T ) ∈ A′ stand for the Kontsevich integral of thecomplement closed in this way.

Here is a figure illustrating the closure operation for tangles:

.

As you can see, the closure is performed by adding the arc .

8.7.6. Lemma. For a one-string tangle T , the following identities hold:

(1) Z(T ) = Z(T ).

(2) Z(K \ T ) = Z(K \ T ).

Proof. (1) Indeed, by Lemma 8.7.5,

Z(T ) = Z

(T

)= Z(T ) · Z

( ).

But each chord diagram on the complement to the tangle is equal to zeroby one-term relation except for the diagram of order 0 (without any chords)which closes the chord diagrams on T into chord diagrams on the circle.

(2) The same argument holds for the complements, too, but it is evensimpler, because in this case on the additional arc there is only one (empty)chord diagram.

8.8. Proof of the Kontsevich theorem 211

8.7.7. Lemma. For a one-string tangle T and its complement K \ T the

concatenation product A′T×A′K\T → A

′agrees with the product in the algebra

A′ in the sense that the following identity holds:

Z(T ) · Z(K \ T ) = Z(T ) · Z(K \ T ).

Proof. This follows directly from the definitions.

Proof of the theorem. By Lemmas 8.7.6 and 8.7.7 we have:

Z

( )= Z

( )· Z( )

= Z

( )· Z( )

.

The theorem is proved.

8.7.8. Definition. The theorem allows one to define the universal Vassilievinvariant by either

I(K) =Z(K)

Z(H)c/2

or

I ′(K) =Z(K)

Z(H)c/2−1,

where c denotes the number of critical points of K in an arbitrary Morse

representation, and the quotient means division in the algebra A′: (1 +a)−1 = 1− a+ a2 − a3 + . . . .

The invariance of Z(K) in the class of Morse knots and the theoremabove imply that both I(K) and I ′(K) are invariant under an arbitrarydeformation of K.

The version I ′(K) is preferable, because it is multiplicative. However,the version I(K) is also used, because it gives sense to the expression “Kont-sevich’s integral of the unknot” (see 1).

8.8. Proof of the Kontsevich theorem

First of all we reformulate the Kontsevich theorem (or, more exactly, theKontsevich part of the Vassiliev–Kontsevich theorem 4.2.1) as follows.

8.8.1. Theorem. Let w be an unframed weight system of order n (i. e.vanishing on chord diagrams whose number of chords is different from n).Then there exists a Vassiliev invariant of order ≤ n whose symbol is w.


Proof. The desired knot invariant is given by the formula

K 7−→ w(I(K)).

Let D be a chord diagram of order n and KD be a singular knot withchord diagram D. The theorem follows from the fact that I(KD) = D +(terms of order > n). Since the denominator of I(K) starts with the unit ofthe algebra A′, it is sufficient to prove that

Z(KD) = D + (terms of order > n)

By definition, Z(KD) is the alternating sum of 2n values of Z on thecomplete resolutions of the singular knot KD. To see what happens at asingle double point let us consider the difference of Z on two knots K+ andK− where an overcrossing in K+ is changed to an undercrossing in K−. Bya horizontal deformation we have:

Z(K+) − Z(K−) = Z

( )−Z( )

= Z

( )−Z( )

.

These two knots differ only in that the first knot has one string that makes asmall curl around another string. We assume that this curl is very thin andcontained in a slice of thickness δ. Enclose the curl in a box of height δ andconsider it as a tangle T+. Let T− be the similar tangle for K− consistingof two parallel vertical strings:

T+ = T− = .

By Lemma 8.7.5 we have

Z(K+) = Z(T+) ·Z(K+ \ T+) Z(K−) = Z(T−) ·Z(K− \ T−) .

But K+ \ T+ coincides with K− \ T−, therefore,

Z(K+)− Z(K−) = (Z(T+)− Z(T−)) · Z(K+ \ T+) .

Now, for the tangle T− both functions z(t) and z′(t) are constant, hence

Z(T−) consists of only one summand: the trivial tangled chord diagram D(0)−

on T− (here ‘trivial’ means with an empty set of chords) with coefficient one.

The zero degree term D(0)+ of Z(T+) is also the trivial chord diagram on T+

with coefficient one. ButD(0)− = D

(0)+ as chord diagrams. Thus the difference

Z(T+)−Z(T−), and therefore the difference Z(K+)−Z(K−), starts with afirst degree term, which comes from the chord connecting the two strings ofT+.

Now consider n double points p1, . . . , pn of the knot KD. Let ε1, . . . , εnbe a sequence of signs ‘+’ or ‘−’. Denote by Kε1,...,εn the knot obtainedfrom KD by a positive (like in K+ above) resolution of pj if εj = + or by anegative (like in K− above) resolution of pj if εj = −. Deforming all knots

8.8. Proof of the Kontsevich theorem 213

Kε1,...,εn near the points p1, . . . , pn in the same manner as above for K+ andK−, we get, for every j = 1, . . . , n, two tangles Tj,+ and Tj,− contained ina box of height δ near the point pj . The tangle Tj,− always consists of twovertical strings. Then we can write

Z(KD) =∑

ε1,...,εn

ε1 · . . . · εn · Z(Kε1,...,εn)

=∑

ε1,...,εn−1

ε1 · . . . · εn−1 · (Z(Tn,+)− Z(Tn,−)) · Z(Kε1,...,εn−1,+ \ Tn,+)

=∑

ε1,...,εn−2

ε1 · . . . · εn−2 · (Z(Tn,+)− Z(Tn,−)) · (Z(Tn−1,+)− Z(Tn−1,−))

· Z(Kε1,...,εn−2,+,+ \ (Tn,+ ∪ Tn−1,+))

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

= (Z(T1,+)− Z(T1,−)) · . . . · (Z(Tn,+)− Z(Tn,−)) · Z(K+,...,+ \ (⋃

j

Tj,+)) .

As above, each difference Z(Tj,+) − Z(Tj,−) starts with a term of degree1. Hence the product Z(KD) starts from degree n. Moreover, the termof degree n is proportional to a chord diagram whose j-th chord connectsthe two strings of the tangle Tj,+. Since the two strings of Tj,+ correspondto the double point pj in KD, this chord diagram is precisely D. Now tocalculate the order n part of Z(KD) we must compute the coefficient of D.It is equal to the product of coefficients of one chord terms in Z(Tj,+). Wewill show that all of them are equal to 1. Indeed, it is possible to choosethe coordinates z and t is such a way that z′(t) ≡ 0 for one string, and foranother one, the point z(t) makes one complete turn around zero when tvaries from 0 to δ:

t

δ

0

z(t)


So we have

1

2πi

δ∫

0

dz − dz′z − z′ =

1

2πi

∮

|z|=1

dz

z= 1

by the Cauchy theorem. The Kontsevich theorem is proved.

In fact, according to the exercise 8.6.7

Z(Tj,+) = exp( )

,

but we actually did not need this explicit result in the previous argument.

We will now state some properties of the universal Vassiliev invariantI(K) which reveal the precise meaning of the word “universal”. The first ofthese properties follows from the argument used in the proof of the Kontse-vich’s theorem above.

8.8.2. Proposition. If In(K) denotes the n-th graded component of theseries I(K), then the function K 7→ In(K) is a Vassiliev invariant of ordern.

The converse assertion is contained in the following

8.8.3. Proposition. Any Vassiliev invariant can be factored through I: for

any v ∈ V there exists a linear function f on A′ such that v = f I.

Proof. Let v ∈ Vn. By Kontsevich’s theorem we know that there is afunction f0 such that v and f0 In have the same symbol. Therefore, thedifference v− f0 In belongs to Vn−1 and is thus representable as f1 In−1.Proceeding in this way, we will finally obtain:

v =n∑

i=1

fi In−i.

Remark. The construction of the previous proof shows that the universalVassiliev invariant induces a splitting of the filtered space V into a directsum with summands isomorphic to the factors Vn/Vn−1. Elements of thesesubspaces are referred to as canonical Vassiliev invariants. We will speakabout them in more detail later in Section 10.4.

The two last propositions, taken together, imply the following theorem.

8.8.4. Theorem. The universal Vassiliev invariant I is exactly as strongas the set of all Vassiliev invariants: for any two knots K1 and K2 we have

I(K1) = I(K2) ⇐⇒ ∀v ∈ V v(K1) = v(K2).

8.9. Group-like property of the Kontsevich integral 215

8.9. Group-like property of the Kontsevich integral

8.9.1. Theorem. For any Morse knot K the Kontsevich integral Z(K) is

a group-like element of the Hopf algebra A′, i.e.

(1) δ(Z(K)) = Z(K)⊗ Z(K) ,

where δ is the comultiplication defined in Section 4.5.4.

Proof. Since the group-like elements form a multiplicative group, it is suf-ficient to prove this property for the Kontsevich integral of a tangle. If Tembedded into a slice Cz × [a, b] ⊂ R3 without maximum and minimumpoints.

In right-hand side of (1), consider the coefficient of the tensor productof two (tangled) chord diagrams D1⊗D2 with m and n chords respectively.It comes from a particular choice of the pairing P1 for m chords of D1

on the levels t1, . . . tm, and a pairing P2 for n chords of D2 on the levelstm+1, . . . , tm+n. Denote by ∆1 the simplex a < tm < · · · < t1 < b, and by∆2 the simplex a < tm+n < · · · < tm+1 < b. Then the coefficient at D1⊗D2

in the right-hand side of (1) is the product of two integrals

(−1)↓1+↓2

(2πi)m+n

∫

∆1

m∧

j=1


·

∫

∆2

m+n∧

j=m+1


,

which can be written as a single integral over the product of simplices:

(−1)↓1+↓2

(2πi)m+n

∫

∆1×∆2

m+n∧

j=1


.

Now we split the product ∆1 ×∆2 into the union of mutually disjoint sim-plices corresponding to all possible shuffles of two linearly ordered wordstm < · · · < t1 and tm+n < · · · < tm+1. A shuffle of two words tm . . . t1 andtm+n . . . tm+1 is a word consisting of the letters tm+n, . . . , t1 and such thatits subwords consisting of letters tm . . . t1 and tm+n . . . tm+1 respect theirlinear orders. Here is an example (m = 2, n = 1) of such splitting:

∆1

-∆2

∆1 ×∆2

6t3

-t1

t2

=

t3 < t2 < t1

⋃

t2 < t3 < t1

⋃

t2 < t1 < t3


The integral over the product of simplices is equal to the sum of integralscorresponding to all possible shuffles. But the integral over one particularsimplex is precisely the coefficient in Z(K) of the chord diagram obtainedby merging the chord diagrams D1 and D2 according to the shuffle. This isequal to one term of the coefficient of D1 ⊗D2 in the left-hand side of (1).It is easy to see that the terms in the coefficient of D1 ⊗D2 in δ(Z(K)) arein one to one correspondence with all ways to merge D1 and D2, or in otherwords, with all possible shuffles of the words tm . . . t1 and tm+n . . . tm+1.

Exercises

(1) Is it true that Z(H) = Z(H), where H is the hump as shown in page193 and H is the same hump reflected in a horizontal line?

(2) M. Kontsevich in his pioneering paper [Kon1] and some of his followers(for example, [BN1, CD3]) defined the Kontsevich integral slightlydifferently, numbering the chords upwards. Namely, ZKont(K) =

=

∞∑

m=0

1

(2πi)m

∫

tmin<t1<···<tm<tmaxtj are noncritical

∑

P=(zj ,z′j)(−1)↓PDP

m∧

j=1


.

Prove that for any tangle T , ZKont(T ) = Z(T ), as series of T -chorddiagrams.

Hint. Change of variables in multiple integrals.

(3) Let BTε be a braiding tangle withthe specific distances between itsendpoints. Prove that

limε→0

Z(BTε) = R⊗ id ,

BTε =

1

ε

where the right-hand side is a series of tangled chord diagrams on BTεwith R from 8.6.7 on the left two strings and free right string.

(4) The Euler dilogarithm is defined by the power series Li2(z) =∞∑

k=1

zk

k2for

|z| ≤ 1. Prove the following identities

Li2(0) = 0; Li2(1) = π2

6 ; Li′2(z) = − ln(1−z)z ;

ddz

(Li2(1− z) + Li2(z) + ln z ln(1− z)

)= 0 ;

Li2(1− z) + Li2(z) + ln z ln(1− z) = π2

6 .

About these and other remarkable properties of Li2(z) see [Lew, Kir,Zag2].

Exercises 217

(5) Consider an associativity tangle shownin details on the right picture. ComputeZ( )

up to the second order.

Answer. − 12πi ln

(1−εε

) (−

)

− 18π2 ln2

(1−εε

) (+

)

0 ε 1−ε 1 z

ε

1−ε

tz=t

+ 14π2

(ln(1− ε) ln

(1−εε

)+ Li2(1− ε)− Li2(ε)

)

− 14π2

(ln(ε) ln

(1−εε

)+ Li2(1− ε)− Li2(ε)

)

The calculation here follows the example 8.7.3 and uses the dilog-arithm function from problem 4. Note that the Kontsevich integraldiverges as ε→ 0.

(6) Make the similar computation Z( )

for the reflected tangle. Describethe difference with the answer to the previous problem.

(7) Compute the Kontsevich integral Z( )

of themaximum tangle shown on the right picture.Answer. + 1

2πi ln(1− ε)+ 1

4π2

(Li2(

ε2−ε)− Li2

( −ε2−ε))

0 1−ε 1 z

1−ε

1

t t=−z2+(2−ε)z

+ 18π2

(ln2 2− ln2

(1−ε2−ε

)+ 2Li2

(12

)− 2Li2

(1−ε2−ε))

+ 18π2

(ln2 2− ln2(2− ε) + 2Li2

(12

)− 2Li2

(1

2−ε))

(8) Compute the Kontsevich integral Z( )

of theminimum tangle shown on the right picture.Answer. − 1

2πi ln(1− ε)+ 1

4π2

(Li2(

ε2−ε)− Li2

( −ε2−ε))

0 ε 1 z

ε

t t=z2−εz+ε

+ 18π2

(ln2 2− ln2

(1−ε2−ε

)+ 2Li2

(12

)− 2Li2

(1−ε2−ε))

+ 18π2

(ln2 2− ln2(2− ε) + 2Li2

(12

)− 2Li2

(1

2−ε))

Note that all nontrivial terms in the last two problems tend to zeroas ε → 0. This is not surprising because they always contain a ‘long’chord.

(9) Obtain the Kontsevich integral of the hump from example 8.7.3 as prod-uct of tangled chord diagrams from problems 5, 7, 8:

Z( )

= Z( )

· Z( )

· Z( )

.

To do this introduce shorthand notation for the coefficients:Z( )

= + A + B + C + D

Z( )

= + E(

−)

+ F(

+)

+ G + H


Z( )

= + I + J + K + L .Show that the order 1 terms of the product vanish.The only nonzero chord diagram of order 2 on the hump is the cross

(diagram without isolated chords). The coefficient of this diagram isB +D +G+ J + L−AE +AI + EI. Show that it is equal to

Li2(

ε2−ε

)−Li2

(−ε2−ε

)+Li2

(12

)−Li2

(1

2−ε

)−Li2(ε)

2π2 + ln2 2−ln2(2−ε)4π2 + 1

24 .

Using the properties of the dilogarithm mentioned in problem 4 provethat the last expression equals 1

24 . This is also a consequence of theremarkable Roger five-term relation (see, for example, [Kir])

Li2x+ Li2y − Li2xy = Li2x(1−y)1−xy + Li2

y(1−x)1−xy + ln (1−x)

1−xy ln (1−y)1−xy

and the Landen connection formula (see, for example, [Roos])

Li2z + Li2−z1−z = −1

2 ln2(1− z) .(10) Let Si be the operation of reversing the orientation of the i-th string of

a tangle T . Denote by the same symbol Si the operation on T -chorddiagrams which multiplies a chord diagram by (−1) raised to the powerof the number of endpoints of chords lying on the i-th string (see theformal definition in 12.3.6). Prove that

Z(Si(T )) = Si(Z(T ))

We shall use this operation in the next chapter.

(11) Compute the Kontsevich integralZ(AT tb,w) up to the order 2. Hereε is a small parameter, and w, t,b are natural numbers subject toinequalities w < b and w < t.Answer. Z(AT tb,w) = +

AT tb,w =

εb

εw

εt

+ 12πi ln

(εw−εt

εb

)− 1

2πi ln(εw−εb

εt

)

− 18π2 ln2

(εw−εt

εb

)− 1

8π2 ln2(εw−εb

εt

)

− 14π2

(ln(εb−w) ln

(εw−εb

εt

)+ Li2(1− εb−w)− Li2(ε

t−w))

+ 14π2

(ln(1− εt−w) ln

(εw−εb

εt

)+ Li2(1− εb−w)− Li2(ε

t−w))

.

(12) How will the Kontsevich’s integral change, if in its definition integrationover simplices is replaced by integration over cubes?

(13) Prove that

Φ = limε→0

ε−w

2πi·(

+)ε−

t2πi· ·Z(AT tb,w)·ε b

2πi· ·ε w

2πi·(

+).

Exercises 219

(14) Prove that for the tangle T tm,b,w on theright picture

limε→0

ε−b−w2πi

(+

)· ε− t−w

2πi ··Z(T tm,b,w)· εb

εw

εm

εt

·εm−w2πi · ε b−w

2πi

(+

)= Φ⊗ id.

(15) Prove that for the tangle T t,mb,w on the

right picture

limε→0

ε−t−w2πi

(+

)· ε−m−w

2πi ··Z(T t,mb,w )· εb

εw

εm

εt

·ε b−w2πi · ε t−w

2πi

(+

)= id⊗ Φ.

Chapter 9

Operations on knots

and Kontsevich’s

integral

We saw in Sec. 8.6.8 that the Kontsevich integral is multiplicative withrespect to the connected sum of knots. In this chapter we consider otheroperations on knots and describe their effect on the Kontsevich integral.The operations in question are:

• R— mirror reflection (changing the orientation of the ambient space),

• S — changing the orientation of a knot,

• MT — mutation of a knot with respect to a distinguished tangle T ,

• ψn — n-th cabling of a knot.

9.1. Reality of the integral

Choose a countable family of chord diagrams Di that constitute a basis inthe linear space A′. The Kontsevich integral of a knot K can be written asan infinite series Z(K) =

∑ciDi whose coefficients ci depend on the knot

K. A priori these coefficients are complex numbers.

9.1.1. Theorem. All coefficients ci of the Kontsevich integral Z(K) =∑ciDi are real numbers.

Remark. Of course, this fact is a consequence of the Le–Murakamitheorem 12.4.12 stating that these coefficients are rational. We state theprevious theorem because it has a simple independent proof which is quiteinstructive.

221

222 9. Operations on knots and Kontsevich’s integral

Proof. Rotate the knot K around the real axis x through 180:

t

t2

t1

0

y

x

Cz

K

t⋆

t⋆2t⋆1

0y⋆

x⋆

Cz⋆

K⋆

t⋆2 = −t1t⋆1 = −t2

and denote the obtained knot by K⋆. To distinguish the objects (space,coordinates etc.) related to the knot K⋆, we will tag the correspondingsymbols with a star. Coordinates t⋆, z⋆ in the space that contains K⋆ arerelated to the coordinates t, z in the ambient space of K as follows: t⋆ = −t,z⋆ = z.

The rotation can be realized by a smooth isotopy, hence the universal

Vassiliev invariants of K and K⋆ coincide: Z(K) = Z(K⋆). The two knotshave the same number c of critical points of the height function. Therefore

Z(K) = Z(K) · Z(h)c/2 = Z(K⋆) · Z(h)c/2 = Z(K⋆),

where h = is the plane unknot with one hump. Therefore, to prove

the reality of Z(K), it is enough to prove that each coefficient of Z(K⋆) isthe complex conjugate to the corresponding coefficient of Z(K).

Fix the number of chords m. Then each pairing P = (zj , z′j), 1 ≤ j ≤m for the knot K corresponds to a pairing P ⋆ = (z⋆j , z⋆j ′) for the knot

K⋆, where z⋆j = zm−j+1 and z⋆j′ = z′m−j+1. Note that the corresponding

chord diagrams are equal: DP = DP ⋆ . Moreover, ↓⋆= 2m−↓ and hence(−1)↓

⋆= (−1)↓. The simplex ∆ = tmin < t1 < · · · < tm < tmax for the

variables ti corresponds to the simplex ∆⋆ = −tmax < t⋆m < · · · < t⋆1 < −tmin

for the variables t⋆i . The coefficient of DP ⋆ in Z(K⋆) is

c(DP ⋆) =(−1)↓

(2πi)m

∫ m∧

j=1

d ln(z⋆j − z⋆j ′) ,

where z⋆j and z⋆j′ are understood as functions in t⋆1, . . . , t⋆m and the integral

is taken over a connected component in the simplex ∆⋆. In the last integralwe make the change of variables according to the formula t⋆j = −tm−j+1.

9.2. Mirror reflection 223

The Jacobian of this transformation is equal to (−1)m(m+1)/2. Therefore,

c(DP ⋆) =(−1)↓

(2πi)m

∫(−1)m(m+1)/2

m∧

j=1

d ln(zm−j+1 − z′m−j+1)

(integral over the corresponding connected component in the simplex ∆).Now permute the differentials to arrange the subscripts in the increasing or-der. The sign of this permutation is (−1)m(m−1)/2. Note that (−1)m(m+1)/2 ·(−1)m(m−1)/2 = (−1)m. Hence,

c(DP ⋆) =(−1)↓

(2πi)m(−1)m

∫ m∧

j=1

d ln(zj − z′j)

=(−1)↓

(2πi)m

∫ m∧

j=1

d ln(zj − z′j) = c(DP ).


9.2. Mirror reflection

Let R be the operation sending a knot to its mirror image. Define the corre-sponding operation R on chord diagrams to be identical on the diagrams ofeven order and to be multiplication by (−1) on the diagrams of odd order.

9.2.1. Theorem. The Kontsevich integral commutes with the operation R:

Z(R(K)) = R(Z(K)) ,

where by R(Z(K)) we mean simultaneous application of R to all the chorddiagrams participating in Z(K).

Proof. Let us realize the operation R on knots by the reflection of R3 =Rt × Cz coming from the complex conjugation in Cz: (t, z) 7→ (t, z).

Consider the Kontsevich integral of K:

Z(K) =∞∑

m=0

1

(2πi)m

∫ ∑

P

(−1)↓DP

m∧

j=1

d ln(zj − z′j) .


Then the Kontsevich integral for R(K) will look as

Z(R(K)) =∞∑

m=0

1

(2πi)m

∫ ∑

P

(−1)↓DP

m∧

j=1

d ln(zj − z′j)

=

∞∑

m=0

1

(2πi)m

∫ ∑

P

(−1)↓DP

m∧

j=1

d ln(zj − z′j)

=∞∑

m=0

1

(2πi)m

∫ ∑

P

(−1)↓DP

m∧

j=1

d ln(zj − z′j) .

We see that the terms of Z(R(K)) with an even number of chords coincide

with those of Z(K) and terms of Z(R(K)) with an odd number of chords

differ from the corresponding terms of Z(K) by a sign. Since Z(K) is real,this implies the theorem.

Remark. Taking into account the fact that the Kontsevich integral isequivalent to the totality of all finite type invariants, Theorem 9.2.1 canbe restated as follows: Let v be an invariant of finite degree n, let K be asingular knot with n double points and K ′ its mirror image. Then v(K) =v(K ′) for even n and v(K) = −v(K ′) for odd n. This fact was first provedby V. Vassiliev in [Va1].

Recall (see p. 20) that a knot K is called amphicheiral, if it is equivalentto its mirror image: K = R(K).

9.2.2. Corollary. The Kontsevich integral Z(K) and the universal Vas-siliev invariant I(K) of an amphicheiral knot K consist only of even orderterms.

Proof. Indeed, the theorem implies that Z(K) = R(Z(K)), hence all theodd order terms in the series Z(K) vanish. The quotient of two even series

in the graded completion A′ is obviously even, therefore the same propertyholds for I(K) = Z(K)/Z(H)c/2, too.

9.3. Orientations of knots

Let S be the operation on knots which inverts their orientation. The sameletter will also denote the analogous operation on chord diagrams (invertingthe orientation of the outer circle or, which is the same thing, axial symmetryof the diagram).

9.4. Mutation 225

9.3.1. Theorem. The Kontsevich integral commutes with the operation Sof inverting the orientation:

Z(S(K)) = S(Z(K)).

Proof. The required identity follows directly from the definition of theKontsevich integral in Sec. 8.2. Indeed, the number ↓ for the knot S(K)with inverse orientation is equal to the number ↑ for the initial knot K.Here by ↑ we of course mean the number of points (zj , tj) or (z′j , tj) in apairing P where the coordinate t increases along the orientation of K. Sincethe number of points in a pairing is always even, (−1)↓ = (−1)↑. Hence thecorresponding chord diagrams appear in Z(S(K)) and in S(Z(K)) with thesame sign. The theorem is proved.

Corollary. The following two assertions are equivalent:

• Vassiliev invariants do not distinguish the orientation of knots,

• all chord diagrams are symmetric: D = S(D) modulo one- andfour-term relations.

The calculations of [Kn0] show that up to order 12 all chord diagramsare symmetric. For bigger orders the problem is still open.

9.3.2. Exercise. Prove the equivalence of the two claims:

• all chord diagrams are symmetric modulo one- and four-term rela-tions.

• all chord diagrams are symmetric modulo only four-term relations.

9.4. Mutation

The purpose of this subsection is to show that the Kontsevich integral com-mutes with the operation of mutation (this fact was first noticed by T. Le).As an application, we will construct a counterexample to the intersectiongraph conjecture (p. 87).

9.4.1. Mutation of knots.

9.4.2. Definition. Suppose we have a knot K with a distinguished tangleT which has two strings at the bottom and two strings at the top. Let uscut out the tangle, rotate it through 180 around a vertical axis and insert itback. This operation MT is called mutation and the obtained knot MT (K)is called a mutant of K.


Here is a widely known pair of mutant knots, the Conway and Kinoshita–Teresaka knots:

C = KT =

9.4.3. Theorem. [MC]. There exists a Vassiliev invariant v of order 11such that v(C) 6= v(KT ).

Morton and Cromwell manufactured the invariant v using the Lie algebraglN with a nonstandard representation (or, in other words, the HOMFLYpolynomial of certain cablings of the knots).

9.4.4. Mutation of the Kontsevich integral. Let K be a knot with adistinguished tangle T which has two strings at the bottom and two stringsat the top. Let MT (K) be the mutant of K obtained by the rotation ofT . We shall define the mutation MT (Z(K)) of the Kontsevich integral asthe simultaneous mutation of all chord diagrams participating in Z(K) (seedefinition below). Then the following theorem holds.

9.4.5. Theorem. [Le]. The Kontsevich integral commutes with the muta-tion operation MT :

Z(MT (K)) = MT (Z(K)).

By lemma 8.7.2 Z(K) does not contain any chord diagram with a chordconnecting T with the remaining part of K. This means that all chorddiagrams which appear in Z(K) are products (in the sense of section 8.7) oftangled chord diagrams on T and chord diagrams on K\T . A chord diagramon the tangle, considered as a part of the whole chord diagram for the knot,will be called a share. In purely combinatorial terms, this definition can bestated as follows.

9.4.6. Definition. A share is a part of the chord diagram consisting of twoarcs of the outer circle and all chords whose endpoints belong to these arcs.Moreover, we require that the whole chord diagram contains no chords thatconnect these two arcs with the remaining part of the circle.

Examples. In the left picture below the collection of four chords insidethe dotted oval is a share. But the collection of three fat chords is not ashare because there exists a fourth chord that separates their ends. In the

9.5. Cablings 227

right picture there are two shares shown by dotted arcs.

&%'$rr

rrr rr rrr rr &%

'$rrr r

r rr r rr r rrrThe choice of a tangle T in a knot K distinguishes the T -shares (which

correspond to the tangled chord diagrams of Z(T )) in all chord diagramsthat appear in Z(K).

9.4.7. Definition. The mutation MT (Z(K)) of Z(K) is the simultaneousrotation of the T -shares in all diagrams of Z(K).

Mutants of the chord diagrams of last two examples look as follows

&%'$rr

rrr r

rrrr rr &%

'$rrr r

r rrrr rr rr r

9.4.8. Proof of the theorem. Let Tr be a tangle obtained from T byrotation under the mutation MT . Then Z(K) = Z(T ) · Z(K \ T ) andZ(MT (K)) = Z(Tr) · Z(K \ T ). So

MT (Z(K)) = MT (Z(T ) · Z(K \ T )) = MT (Z(T )) · Z(K \ T )

= Z(Tr) · Z(K \ T ) = Z(MT (K)) .


9.4.9. Counterexample to the Intersection Graph Conjecture. Itis easy to see that the mutation of chord diagrams does not change theintersection graph. So, if the intersection graph conjecture (see 4.3.3) weretrue, the Kontsevich integrals of mutant knots would coincide. Hence allVassiliev invariants would take the same value on mutant knots. But thiscontradicts Theorem 9.4.3. Therefore the intersection graph conjecture isfalse.

9.5. Cablings

9.5.1. Cablings of knots. Cablings are defined for framed knots in thefollowing way. The framing, as a section of the normal bundle of a knotK, determines a trivialization of the bundle and supplies it with a com-plex structure, where the framing vector at every point ends at 1 of thecorresponding normal plane considered as the plane of complex numbers.

The n-th cabling ψn(K) is a knot consisting of a bunch of n strings thatfollow the knot K in a narrow toroidal neighborhood rotating with respect


to the framing so that after one full turn around K they close up with aclockwise rotation by 2π/n. .

ψ4(K)

K

4-th cabling of K

ψ4(K)

K

4-th cabling with respect

to the blackboard framing

In formulas this definition can be expressed as follows. Consider K asthe image of a map K : S1 → R3, where S1 is the unit circle in the complexplane S1 = z ∈ C : |z| = 1. Let f(z) be the framing. Then we set

ψn(K)(z) := K(zn) + εzf(zn) ,

where ε is a small positive number and operations are understood in termsof the complex structure on normal planes.

The knot ψn(K) inherits the framing of K.

9.5.2. Cablings of knot invariants. Let v be a knot invariant. Its n-thcabling ψ∗nv is defined in a natural way by setting

ψ∗nv(K) = v(ψn(K)) .

9.5.3. Proposition. If v is a framed Vassiliev invariant of order 6 m, thenψ∗nv is a Vassiliev invariant of order 6 m too.

Proof. We prove the proposition only for n = 2; for arbitrary n the proofis similar and we leave it to the reader.

According to Vassiliev’s skein relation (Equation (1) on page 59), theextension of ψ∗2v to singular knots satisfies

ψ∗2v( ) = ψ∗2v( )− ψ∗2v( )

= v( )− v( )

9.5. Cablings 229

= v( ) + v( ) + v( ) + v( ) .

Hence the vanishing of v on knots with more than m double points impliesthe vanishing of ψ∗2v on knots with more than m double points.

9.5.4. Cablings of chord diagrams. In this section we explore how thesymbols of Vassiliev invariants v and ψ∗nv are related to each other.

For a chord diagram D define ψn(D) to be the sum of chord diagramsobtained by all possible ways of lifting the ends of the chords to the n sheetedcovering of the circle of D. We extend ψn to the space spanned by chorddiagrams by linearity.

Examples.

ψ2( ) = + + + + +

+ + + + + +

+ + + +

= 12 + 4 .

ψ2( ) = 8 + 8 .

It is easy to see that the mapping ψn satisfies the 4-term relation (Exer-cise 1 at the end of the Chapter), so it descends to the graded space A. Theexamples above show that it is not an algebra homomorphism. However, itis a coalgebra automorphism according to Exercise 7 below.

Theorem. Let v be a Vassiliev invariant. Then

symb(ψ∗nv)(D) = symb(v)(ψn(D)) .

The proof follows directly from a careful analysis of the proof of Propo-sition 9.5.3.

9.5.5. Cablings of closed and open Jacobi diagrams. Since the al-gebra A is isomorphic both to the algebra of closed diagrams C (Theorem5.3.1) and to the algebra of open diagrams B (Section 5.7), the operationψn transfers to these spaces, too. Below, we give an explicit description ofthe resulting operators.


Proposition. For a closed diagram C, ψn(C) is equal to the sum of closeddiagrams obtained by all possible ways of lifting the univalent vertices of Cto the n sheeted covering of the Wilson loop of C.

Proof. Induction on the number of internal vertices of C. If there are nointernal vertices, then C is a chord diagram and the claim to be provedcoicides with the initial definition of ψn.

In general we may use STU relation to decrease the number of internalvertices and then use the induction hypothesis. The only thing one shouldcheck is that ψn is compatible with the STU relation. We show how it worksfor n = 2:

ψ2

( )= ψ2

( )− ψ2

( )

= + + +

− − − −

= − + −

= + .

9.5.6. Corollary. [KSA] Every open diagram B with k univalent vertices(legs) is an eigenvector of the linear operator ψn with eigenvalue nk.

Proof. The isomorphism χ : B ∼= C takes an open diagram B ∈ B withk legs into the average of the k! closed diagrams obtained by all possiblenumberings of the legs and attaching the Wilson loop according to the num-bering. The value of ψn on each closed diagram is equal to the sum of the nk

diagrams obtained by all possible lifts of its legs to the n-sheeted coveringof the Wilson loop. Therefore, ψn(B) is 1/k! times the sum of nkk! closeddiagrams arranged in a 2-dimensional array with nk columns and k! rows.Each column, corresponding to a specific lift to the covering, contains thediagrams that differ from one another by all possible renumberings of theirlegs. The sum over each column divided by k! is thus equal to B. As thenumber of columns is nk, we obtain ψn(B) = nkB.

9.5. Cablings 231

9.5.7. Cablings of the Lie algebra weight systems. Given a semi-simple Lie algebra g, in Chapter 7.1 we constructed the universal Lie algebraweight system ϕg : A → U(g). If, additionally, a representation V of g isspecified, then we have a numeric weight system ϕVg : A → C. The cablingoperation ψn acts on these weight systems as follows:

ψ∗nϕg(D) := ϕg(ψn(D)), ψ∗nϕVg (D) := ϕVg (ψn(D))

for any chord diagram D.

The construction of the universal Lie algebra weight system (Section7.1.1 rests on the assignment of basic vectors ei ∈ g to the endpoints ofi’th chord, then taking their product along the Wilson loop and summingup over each index i. For the weight system ψ∗nϕg, to each endpoint of achord we not only assign a basic vector, but we also indicate the level ofthe covering to which that particular point is lifted. To form an element ofthe universal enveloping algebra we must read the letters ei along the circlen times. On the first pass we read only those letters which are related tothe first sheet of the covering, omitting all the others. Then read the circlefor the second time and now collect only the letters from the second sheet,etc. up to the n-th reading. The products of ei’s thus formed are summedup over all assignments and over all ways of lifting the endpoints to thecovering.

The weight system ψ∗nϕg can be expressed through ϕg by means of acompact formula. To state it, we need two auxiliary operations in the tensoralgebra of U(g). Let µ : U(g)⊗U(g)→ U(g) and δ : U(g)→ U(g)⊗U(g) bethe multiplication and comultiplication in U(g) (see Section A.1.7). Definethe operations µn : U(g)⊗n → U(g) and δn : U(g)→ U(g)⊗n as compositions

µn := µ (id⊗ µ

) · · ·

((id)⊗(n−2) ⊗ µ

),

δn :=((id)⊗(n−2) ⊗ δ

) · · ·

(id⊗ δ

) δ .

In other words, µn converts tensor products of the elements of U(g) intoordinary products, while δn sends each element g ∈ g into

δn(g) = g ⊗ 1⊗ . . .⊗ 1 + 1⊗ g ⊗ . . .⊗ 1 + · · ·+ 1⊗ 1⊗ . . .⊗ g .9.5.8. Proposition. For a chord diagram D we have

ψ∗nϕg(D) = µn δn(ϕg(D)) .

We leave the proof of this proposition to the reader as an exercise (no.11 at the end of the chapter).

9.5.9. Cabling the weight system ϕVg . Recall that the weight system

ϕVg associated with a representation T : g → V is obtained from ϕg as


follows (see Section 7.1.4 for details). Replace each occurence of ei in ϕg(D)by the corresponding linear operator T (ei) ∈ End(V ) and compute the sumof products of all such operators according to the arrangement of indicesalong the circle. The trace of the obtained operator is ϕVg (D).

Informally speaking, with each end of a chord of D and with an elementei ∈ g assigned to it we accociate a linear operator V → V vizualised by thepicture

ei

V

V

.

To pass to the n-th cabling of ϕVg we must take the sum of n such operators,one per each sheet of the covering:

ei

VV

VV

+

ei

VV

VV

.

Instead of this sum, we may just as well consider a single operator V ⊗n →V ⊗n:

ei

V ⊗ V

V ⊗ V

acting according to the rule

ei(v1 ⊗ v2) = ei(v1)⊗ v2 + v1 ⊗ ei(v2)used to define the n-th tensor power V ⊗n of the representation V (in ourexample n = 2). We think about the tensor factor number i as attachedto the i-th sheet of the covering. These considerations lead to the followingproposition.

More comments will be added!!!

9.5.10. Proposition.

ψ∗nϕVg = ϕV

⊗n

g .

9.5.11. Cabling the Kontsevich integral.

To describe the behavior of the Kontsevich integral with respect to ca-

bling we will need a twisted operation ψn on chord diagrams. Its definitionuses the tangled chord diagram a := ∆n−2

2 (R)∆n−32 (Φ). Here R is the tan-

gled chord diagram on two braided strings introduced in Sec. 8.6.6 and 12.3.5and used throughout Sec. 12.4

9.5. Cablings 233

R = exp

(

2

)· = +

1

2+

1

2 · 22+ . . . ,

Φ is the associator from Sec. 12.3.5, and ∆2 is the operation of doubling thesecond string on tangled chord diagrams introduced in Sec. 12.3.6. Here isan example with n = 3.

a = +1

2

(+

)+

1

8

(+ + +

)

− 1

24

(−

)+ (terms of higher order).

Note that in the case n = 2 the formula for a is simpler, a = R.

In fact a is an infinite series of chord diagrams. It is equal to the combi-natorial Kontsevich integral (Sec. 12.3.5) of the parenthesized tangle whichrepresents the pattern how the n strings of the cable close up into a singlecircle (see the pictures on p. 228).

Now we modify the definition of ψn(D) by insertion of the tangled chorddiagram series a instead of empty (without chords) portion at the placewhere strings go from one sheet of the covering to another one. The result

will be denoted by ψn. The next picture illustrates this notion.

ψ2( ) = 12 + 4 +1

2

(+ + +

+ + + + + + +

+ + + + +)

+ . . .

= 12 + 4 + 2 + 4 + 2 + . . . .

Here, inside the parentheses with common coefficient 1/2 in each of the 16diagrams you can see one and the same fragment with one chord near thecrossing which is precisely the second term in the expansion of R.

The operator ψn maps an individual chord diagram D not into a finitecombination of chord diagram of the same degree as ψn does, but ratherinto an infinite series of chord diagrams whose lowest part is just ψn(D).

Theorem. For a framed knot K

Ifr(ψn(K)) = ψn(Ifr(K)) ,


where Ifr is the framed version of the Kontsevich integral from 10.1, andthe cabling operation ψn on the right-hand side is applied to every chorddiagram of the series Ifr(K).

Proof. The theorem directly follows from Le-Murakami’s approach to theframed Kontsevich integral (Sec. 10.1.3). For a parenthesized presentationof a knot K with blackboard framing we consider the similar parenthesizedpresentation of ψn(K). All elementary tangles of the last one will be thesame as they were for K only every string for K will be replaced by abunch of n parallel string. The effect of this operation on the combinatorialKontsevich integral of such a tangle is in application of the operations ∆n−1

i

to every string of K. This precisely corresponds to all possible ways of liftingthe ends of the chords to the n sheeted covering. Besides this in ψn(K) wehave one extra elementary tangle located at the place where the n stringsof the cable close up into a single circle. This elementary tangle contributesa := ∆n−2

2 (R)∆n−32 (Φ) into the final combinatorial Kontsevich integral. All

together this sums up to the operation ψn performed on every chord diagramof the final combinatorial Kontsevich integral Ifr(K).

Exercises

(1) (D. Bar-Natan [BN1]). Check that the mapping ψn is compatible thefour-term relation.

(2) Compute ψ3( ) and ψ3( ).

(3) Compute the eigenvalues and eigenvectors of ψ3|A2.

(4) Compute ψ2( ), ψ2( ), ψ2( ), ψ2( ), and ψ2( ).

(5) Compute the eigenvalues and eigenvectors of ψ2|A3.

(6) Compute ψ2(Θm), where Θm is a chord di-

agram with m isolated chords, e.g. the oneshown on the right.

Θm =

m chords

(7) (D. Bar-Natan [BN1]). Prove that the mapping ψn commutes with thecomultiplication of chord diagrams, i.e. the identity

δ(ψn(D)) =∑

J⊆[D]

ψn(DJ)⊗ ψn(DJ)

holds for any chord diagram D.

(8) (D. Bar-Natan [BN1]). Prove that ψm ψn = ψmn.

Exercises 235

(9) Prove the theorem from Section 9.5.4:

symb(ψ∗nv)(D) = symb(v)(ψn(D)) .

(10) Show that the linear mapping ψn of the space B preserves the productin B defined in 5.6 as a disjoint union of graphs. Thus ψn is an auto-morphism of the algebra B and hence an automorphism of one of thetwo Hopf algebra structures on B (see 5.8).

(11) Prove the proposition from Sec.9.5.7:

ψ∗nϕg(D) = µn δn(ϕg(D)) .

Chapter 10

The Kontsevich

integral: advanced

topics

10.1. Framed version of the Kontsevich integral

For framed knots (and links) the Kontsevich integral was first defined byLe and Murakami [LM2] using their combinatorial construction. ThenV. Goryunov defined [Gor1] the Kontsevich integral in an analytic waywhich resembles the original definition from Sec.8.2 and which works equallywell for any framing.

The main distinction of this theory from the one studies in Chapter 8is that the framed version of the Kontsevich integral takes values in thealgebra A of chord diagrams modulo 4-term relations only.

10.1.1. Goryunov’s approach. Let Kε be a copy of K shifted a smalldistance ε in the direction of the framing. We assume that both K and Kε

are in general position with respect to a height function t as in Section 8.2.Then we construct an integral Z(K,Kε) defined by the very same formula

Z(K,Kε) =

∞∑

m=0

1

(2πi)m

∫

tmin<tm<···<t1<tmaxtj are noncritical

∑

P=(zj ,z′j)(−1)↓DP

m∧

j=1


,

where the pairings P = (zj , z′j) and the corresponding chord diagrams DP

are understood in a different way. Namely, we consider only those pairings

237

238 10. The Kontsevich integral: advanced topics

where zj lies on K while z′j lies on Kε. To obtain the chord diagram DP we

consider K as an oriented circle and connect a point (zj , tj) ∈ K on it witha neighbor of the point (z′j , tj) ∈ Kε lying on K. So every time whenever we

choose neighboring points (zj , z′j) in a pairing P we get an isolated chord of

DP . The next picture illustrates these notions.t

K

Kε

t3z3

z′3

t2z′2 z2

t1 z′1z1

DP

z3z′3

z′2

z2

z′1

z1

Now the framed Kontsevich integral may be defined as

Zfr(K) = limε→0

Z(K,Kε) .

In [Gor1] V. Goryunov proved that the limit does exist and invariantunder the deformations of the framed knot K in the class of framed Morseknots. He used [Gor2] this construction to study Arnold’s J+-theory ofplane curves (or equivalently Legendrian knots in solid torus).

Having the definition of Zfr(K) we can define the final framed Kontse-vich integral in a usual way

Ifr(K) =Zfr(K)

Zfr(H)c/2,

where H is the hump unknot from page 193 with the blackboard framing.

The importance of the framed final Kontsevich integral Ifr(K) is thatit establishes the framed version of Theorem 8.8.1 (the Kontsevich part ofthe Vassiliev-Kontsevich theorem 4.2.1):

10.1.2. Theorem. Let w be a framed weight system of order n (i. e. van-ishing on chord diagrams whose number of chords is different from n). Thenthere exists a framed Vassiliev invariant of order ≤ n whose symbol is w.

10.1.3. Le–Murakami’s approach. For the integral expression Zfr(K) =limε→0

Z(K,Kε) one may try to do the combinatorial constructions similar to

those of Sec.12.3.5. It is more convenient to use the blackboard framing.This will lead to the original combinatorial definition of the framed Kontse-vich integral of [LM2]. It turns out that the basic elementary ingredients of

10.3. The Kontsevich integral for braids TBW 239

the construction, the tangled chord diagrams max/min, R, and Φ remainliterally the same as in the unframed case. The whole framed integral isconstructed from them using operations ⊗, Si, ∆i from Sec.12.3.6, and themultiplication of tangled chord diagrams. The only difference in the framedcase is that in the final product of the tangled chord diagrams we shouldtreat the diagrams with isolated chords as non-zero elements of the algebraA.

10.1.4. BGRT’s approach. There is the third way to define the finalframed Kontsevich integral [BGRT]:

Ifr(K) = ew(K)

2Θ · I(K) ∈ A,

where K is a framed knot, K is the same knot without framing, w(K) isthe writhe of K (the linking number of K and a copy of K shifted slightlyin the direction of the framing), Θ is the chord diagram with one chord,and the exponent means the usual power series in the algebra A. Here,

the universal Vassiliev invariant I(K) ∈ A′ is understood as an element ofthe completed algebra A (without 1-term relations) by virtue of a naturalinclusion A′ → A defined as identity on the primitive subspace of A′.

10.1.5. Group-like property. The framed Kontsevich integral Zfr(K)

is a group-like element of the algebra A. This fact is proved in exactlythe same way as the corresponding property of the unframed Kontsevich’sintegral, see Section 8.9.

10.2. Relation with quantum invariants

The quantum knot invariant coresponding to the Lie algebra g and its finite-dimensional representation ρ can be obtained from the Kontsevich integralas follows. Let W V

g (K) be this invariant evaluated on a knot K. It is a

function of the parameter q. Let ϕVg be the weight system defined by the

pair (g, V ) as in 7.1.4. Take the value of ϕVg on the n-th homogeneous termof the Kontsevich invariant I(K), multiply by hn and sum for n from 0 to∞. The resulting function coincides with W V

g (K), where q is replaced by

eh.

10.3. The Kontsevich integral for braids TBW

TBW


10.4. Canonical Vassiliev invariants

The Fundamental Theorem 4.2.1 (more precisely, Theorem 8.8.1) and itsframed version 10.1.2 provide a means to recover a Vassiliev invariant oforder 6 n from its symbol up to invariants of smaller order. So it is natural toconsider those remarkable Vassiliev invariants whose recovery gives preciselythe original invariant.

10.4.1. Definition. ([BNG]) A (framed) Vassiliev invariant v of order6 n is called canonical if for every (framed) knot K,

v(K) = symb(v)(I(K)) .

In the case of framed invariants one should write Ifr(K) instead of I(K).(Recall that I denotes the final Kontsevich integral).

Suppose we have a Vassilev invariant fn of order 6 n for every n. Then

we can construct a formal power series invariant f =∞∑

n=0

fnhn in a formal

parameter h. Let wn = symb(fn) be their symbols. Then the series w =∞∑

n=0

wn ∈ W (see page 96) can be understood as the symbol of the whole

series symb(f).

The series f is called canonical if

f(K) =

∞∑

n=0

(wn(I(K))

)hn ,

for every knot K. And again in the framed case one should use Ifr(K).

Canonical invariants define a grading in the filtered space of Vassilievinvariants which is consistent with the filtration.

Example. The trivial invariant of order 0 which is identically equal to 1on all knots is a canonical invariant. Its weight system is equal to 10 in thenotation of Section 4.6.

!!! tables of canonical invariants up to order 5 will beadded here!!! c2, j3,...

Surprisingly many of the classical knot invariants discussed in Chapters2 and 3 turn out to be canonical.

The notion of a canonical invariant allows to reduce various relationsbetween Vassiliev knot invariants to some combinatorial relations betweentheir symbols, which gives a powerful tool to investigate knot invariants.This approach will be used in section 14.1 to prove the Melvin–Morton

10.4. Canonical Vassiliev invariants 241

conjecture. Now we will give examples of canonical invariants following[BNG].

10.4.2. Quantum invariants. Building on the work of Drinfeld [Dr1,Dr2] and Kohno [Koh2], T. Le and J. Murakami [LM3, Th 10], and C. Kas-sel [Kas, Th XX.8.3] (see also [Oht1, Th 6.14]) proved that the quantumknot invariants θfr(K) and θ(K) introduced in Section 2.6 become canonicalseries after substitution q = eh and expansion into a power series in h.

The initial data for these invariants is a semi-simple Lie algebra g andits finite dimensional irreducible representation Vλ, where λ is its highest

weight. To emphasize this data we shall write θVλg (K) for θ(K) and θfr,Vλ

g (K)

for θfr(K) .

The quadratic Casimir element c (see Section 7.1) acts on Vλ as mul-tiplication by a constant, call it cλ. The relation between the framed andunframed quantum invariants is

θfr,Vλg (K) = q

cλ·w(K)

2 θVλg (K) .

Set q = eh. Write θfr,Vλg and θVλ

g as power series in h:

θfr,Vλg =

∞∑

n=0

θfr,λg,n hn θVλ

g =

∞∑

n=0

θλg,nhn.

According to the Birman–Lin theorem 3.7.6 the coefficients θfr,λg,n and

θλg,n are Vassiliev invariants of order n. The Le–Murakami—Kassel Theoremstates that they both are canonical series.

It is important that the symbol of θfr,Vλg is precisely the weight system

ϕVλg described in Chapter 7. The symbol of θVλ

g equals ϕ′Vλg . In other words,

it is obtained from the previous symbol ϕVλg by the standard deframing

procedure of Sec. 4.6.3. So the knowledge the Kontsevich integral and these

weight systems allows us to restore the quantum invariants θfr,Vλg and θVλ

g

without the quantum procedure of Sec. 2.6.

10.4.3. Colored Jones polynomial. The colored Jones polynomials Jk :=

θVλsl2

and Jfr,k := θfr,Vλsl2

are particular cases of quantum invariants for g = sl2.For this Lie algebra the highest weight is an integer λ = k−1, where k is thedimension of the representation, so in our notation we may use k instead of

λ. The quadratic Casimir number in this case is cλ = k2−12 , and the relation

between the framed and unframed colored Jones polynomials is

Jfr,k(K) = qk2−1

4·w(K)Jk(K) .

The ordinary Jones polynomial of Section 2.4 corresponds to the case k = 2,i.e. to the standard 2-dimensional representation of the Lie algebra sl2.


Set q = eh. Write Jfr,k and Jk as power series in h:

Jfr,k =∞∑

n=0

Jfr,kn hn Jk =∞∑

n=0

Jknhn.

Both series are canonical with the symbols

symb(Jfr,k) = ϕVksl2, symb(Jk) = ϕ′Vk

sl2

defined in Sections 7.1.3 and 7.2.2.

10.4.4. HOMFLY polynomial. This is a particular case of quantum in-variant for the Lie algebra slN and its standard N -dimensional representa-tion Vλ = St. It is known (see, for example, [Oht1, Equations (6.36-38)])that the framed version of this quantum invariant satisfies the followingskein relation

q1

2N θfr,StslN( ) − q−

12N θfr,StslN

( ) = (q1/2 − q−1/2)θfr,StslN( )

and the following framing and initial conditions

θfr,StslN( ) = q

N−1/N2 θfr,StslN

( )

θfr,StslN( ) =

qN/2 − q−N/2q1/2 − q−1/2

.

The quadratic Casimir number in this case is N − 1/N . The deframing

of this invariant gives θStslN= q−

N−1/N2·wθfr,StslN

which satisfies

qN/2θStslN( ) − q−N/2θStslN

( ) = (q1/2 − q−1/2)θStslN( ) ;

θStslN( ) =

qN/2 − q−N/2q1/2 − q−1/2

.

The set of these invariants for all values of N is equivalent to the HOMFLYpolynomial.

After the substitution q = eh and expansion into a power series in h

both θfr,StslNand θStslN

become canonical series. Moreover, their symbols areequal to

symb(θfr,StslN) = ϕStslN

, and symb(θStslN) = ϕ′StslN

,

i.e. precisely those which we considered in Section 7.1.6 (see also the prob-lems (13 – 14) in that chapter).

10.4. Canonical Vassiliev invariants 243

10.4.5. Alexander–Conway polynomial. Consider the unframed quan-tum invariant θStslN

as a function of the parameter N . Let us think about Nnot as a discrete parameter but rather as a continuous variable, where fornon integer N the invariant θStslN

is defined by the skein and initial relations

above. Its symbol ϕ′StslN= ϕ′StglN

(see problem (14) of the Chapter 7) also

makes sense for all real values of N , because for every chord diagram D,ϕ′StglN

(D) is a polynomial of N . Even more, since this polynomial is divisible

by N , we may consider the limit

limN→0

ϕ′StslN

N.

Exercise. Prove that the weight system defined by this limit coincides

with the symbol of the Conway polynomial, symb(C) =∞∑

n=0

symb(cn).

Hint. Use exercise (15) from Chapter 3.

Make the substitution θStslN

∣∣q=eh . The skein and initial relations for θStslN

allow us to show (see exercise (4)) that the limit

A := limN→0

θStslN

∣∣q=eh

N

does exist and satisfies the relations

(1) A( ) − A( ) = (eh/2 − e−h/2)A( ) ;

(2) A( ) =h

eh/2 − e−h/2 .

A comparison of these relations with the defining relation for the Conwaypolynomial 2.3.1 shows that

A =h

eh/2 − e−h/2C∣∣t=eh/2−e−h/2 .

The conclusion is that despite of the Conway polynomial C itself is not acanonical series it becomes canonical after the substitution t = eh/2 − e−h/2and multiplication by h

eh/2−e−h/2 . The weight system of this canonical series

is the same as for the Conway polynomial. Or, in other words,

h

eh/2 − e−h/2C∣∣t=eh/2−e−h/2(K) =

∞∑

n=0

(symb(cn) I(K)

)hn .


Warning. We cannot do the same for framed invariants because none ofthe limits

limN→0

θfr,StslN

∣∣q=eh

N, lim

N→0

ϕStslN

N

exists.

10.5. Wheeling TBW

This is about the wheeling formula — an algebra isomorphism from B toA which is the formal diagrammatic counterpart of the Duflo operator inthe theory of Lie algebras. It will be used in the next section to deduce anexplicite ‘wheels’ formula for the Kontsevich integral of the unknot.

10.6. The Kontsevich integral of the unknot

TBW

The following formula for the framed Kontsevich integral of the unknotwas conjectured in [BGRT] and proved in [BLT] (and also in ...). Let Obe the unknot with zero writhe number.

10.6.1. Theorem.

(1) Ifr(O) = exp∞∑

n=1

b2nw2n = 1 + (∞∑

n=1

b2nw2n) +1

2(∞∑

n=1

b2nw2n)2 + . . .

Here b2n are modified Bernoulli numbers, i.e. the coefficients of theTaylor series

∞∑

n=1

b2nx2n =

1

2lnex/2 − e−x/2

x,

(b2 = 1/48, b4 = −1/5760, b6 = 1/362880,. . . ), and w2n are the wheels, i. e.open diagrams (see 105) of the form

w2 = , w4 = , w6 = , . . .

The sums and products are understood as operations in the algebra of opendiagrams B, which is isomorphic to the algebra of chord diagrams A.

10.7. Rozansky’s rationality conjecture 245

10.7. Rozansky’s rationality conjecture

This section concerns a generalization of the wheels formula for the Kont-sevich integral of the unknot to arbitrary knots. The generalization is,however, not complete – the Rozansky–Kricker theorem only suggests thatIfr(K) has a certain form.

It turns out that the terms of the Kontsevich integral Ifr(K) with valuesin B can be rearranged into lines corresponding to the number of loops inopen diagrams from B. Namely, for any term of Ifr(K), shaving off all legsof the corresponding diagram G ∈ B, we get a 3-graph γ ∈ Γ. Infinitly manyterms of Ifr(K) have the same γ. It turns out that these terms behave in aregular fashion, so that it is possible to recover all of them from γ and somefinite information.

To make this precise we introduce marked open diagrams which are rep-resented by a 3-graph whose edges are marked by power series (it doesmatter on which side of the edge the mark is located, and we will indicatethe side in question by a small leg near the mark). We use such markedopen diagrams to represent infinite series of open diagrams which differ bythe number of legs. More specifically, an edge marked by a power seriesf(x) = c0 + c1x + c2x

2 + c3x3 + . . . means the following series of open

diagrams:

f(x) := c0 + c1 + c2 + c3 + . . .

In this notation the wheels formula 1 can be written as

ln Ifr(O) =

12

ln ex/2−e−x/2

x

Now we can state the

Rozansky’s rationality conjecture. [Roz2]

ln Ifr(K) =

12

ln ex/2−e−x/2

x− 1

2lnAK(ex)

+finite∑

i

pi,1(ex)/AK(ex)

pi,2(ex)/AK(ex)

pi,3(ex)/AK(ex)

+ (terms of (> 3)-loop) ,

where AK(t) is the Alexander polynomial of K normalized so that AK(t) =AK(t−1) and AK(1) = 1, pi,j(t) are polynomials, and the higher loop terms


mean the sum over marked 3-graphs (with finitely many copies of each graph)whose edges are marked by a polynomial in ex divided by AK(t).

The word ‘rationality’ refers to the fact that the labels on all 3-graphsof degree ≥ 1 are rational functions of ex. The conjecture was proved byA. Kricker in [Kri2]. Due to AS and IHX relations the specified presentationof the Kontsevich integral is not unique. So the polynomial pi,j(t) themselvescannot be knot invariats. However, there are certain combinations of thesepolynomials that are genuine knot invariants. For example, consider thepolynomial

Θ′K(t1, t2, t3) =∑

i

pi,1(t1)pi,2(t2)pi,3(t3) ,

Its symmetrization,

ΘK(t1, t2, t3) =∑

ε=±1i,j,k=1,2,3

Θ′K(tεi , tεj , t

εk) ∈ Q[t±1

1 , t±12 , t±1

3 ]/(t1t2t3 = 1) ,

over the order 12 group of symmetries of the theta graph, is a knot invariant.It is called the 2-loop polynomial of K. Its values on knots with few crossingsare tabulated in [Roz2]. T. Ohtsuki [Oht1] found a cabling formula for the2-loop polynomial and its values on torus knots T (p, q).

Exercises

(1) Xheck that c2 and j3 are canonical invariants.

(2) Find a basis in the space of canonicalinvariants of degree 4.

(3) Show that the self-linking number defined in Section 2.2.3 is a canonicalframed Vassiliev invariant of order 1.

(4) Show the existence of the limit from Sec.10.4.5

A = limN→0

θslN ,V

∣∣q=eh

N.

Hint. Use an induction on appropriately chosen complexity of a linkdiagram in such a way that two of the diagrams participating in theskein relation for θslN ,V are strictly simpler then the third one.

(5) Let f(h) =∞∑

n=0

fnhn and g(h) =

∞∑

n=0

gnhn be two power series Vassiliev

invariants, i.e. for every n both fn and gn are Vassiliev invariants oforder 6 n.

Exercises 247

a). Prove that their product f(h) · g(h) as formal power series in h isa Vassiliev power series invariant, and

symb(f · g) = symb(f) · symb(g) .

b). Suppose that f and g are related to each other via a substitutionand a multiplication:

f(h) = β(h) · g(α(h)

),

where α(h) and β(h) are formal power series in h, and

α(h) = ah+ (terms of degree > 2) , β(h) = 1 + (terms of degree > 1) .

Prove that symb(fn) = ansymb(gn).

(6) Prove that a canonical Vassiliev invariant is primitive if its symbol isprimitive.

(7) Prove that the product of any two canonical Vassiliev power series is acanonical Vassiliev power series.

(8) If v is a canonical Vassiliev invariant of odd order andK an amphicheiralknot, then v(K) = 0.

(9) Let κ ∈ Wn be a weight system of degree n. Construct another weightsystem (ψ∗jκ)

′ ∈ W ′n, where ψ∗j is the j-th cabling operator from Section

9.5, and (·)′ is the deframing operator from Section 4.6.3. Thus we havea function fκ(j) : j 7→ (ψ∗jκ)

′ with values in W ′n. Prove that

a). fκ(j) is a polynomial in j of degree 6 n if n is even, and of degree6 n− 1 if n is odd.

b). The n-th degree term of the polynomial fκ(j)

is equal to −κ(wn)2

symb(cn)jn , where wn is

wn =

n spokes

the wheel with n spokes, and cn is the the n-th coefficient of theConway polynomial.

(10) Prove that, after being carried over from B to A, the right hand part ofEquation 1 (page 244) belongs in fact to the subalgebra A′ ⊂ A. Findan explicit expression of the series through some basis of A′ up to degree2.

Answer. The infinite series giving the Kontsevich integral of theunknot, begins as follows:

I(O) = 1− 1

24− 1

5760+

1

1152+

1

2880+ . . .

Note that this agrees well with the answer to Exercise in Section 10.1.3.


(11) Find the framed Kontsevich integrals Zfr( )

and Ifr( )

up to

order 4 for the indicated unknot with blackboard framing.

Answer.

Zfr( )

= 1− 124 + 1

24 + 75760 − 17

5760

+ 72880 − 1

720 + 11920 + 1

5760 .

Ifr( )

= 1/Zfr

( ).

Part 4

Other Topics

Chapter 11

The case of links

— write a couple of pages and insert it to some otherChapter as a section!! —

TBW

11.1. Diagrammatic algebras: general case

11.1.1. Chord diagrams with arbitrary support.

11.1.2. Definition. Let X be an oriented 1-dimensional manifold, i.e. adisjoint union of several oriented real lines and circles. A chord diagramwith support X is a set of pairs of distinct points of X considered up to anorientation- and component-preserving diffeomorphism of X. The space ofchord diagrams on X, denoted by A(X), is the vector space spanned by allchord diagrams modulo the four-term relation.

Let X(m,n) be the manifold consisting of m circles and m lines andA(m,n) = A(X(m,n)). The classical case is A(0, 1) = A. We will also usethe shorthand notation A(m) = A(m, 0). Closing one linear component intoa circle yields a homomorphism ιpq : A(p, q)→ A(p− 1, q + 1).

Theorem. The homomorphism ιpq is an isomorphism if and only if p = 1.In particular, A(1, 0) ∼= A(0, 1) which is a finite type reflection of the factthat the theory of long knots is equivalent to the theory of closed knots.

TO BE FINISHED!!

11.1.3. Jacobi diagrams with structured legs.

251

252 11. The case of links

11.2. Relation to representation theory

11.3. Vassiliev invariants of links

Exercises

(1) Prove that the diagram

1

21 is equal to 0 in the space B(2).

Chapter 12

The Drinfeld associator

The standard (Knizhnik–Zamolodchikov) Drinfeld associator ΦKZ is a veryspecial infinite series in two non-commuting variables whose coefficients arecombinations of multiple zeta values. For us, it is most important that theassociator provides a building block for the construction of the Kontsevichintegral for knots and tangles. Actually, in this construction only some prop-erties of ΦKZ are used; adopting them as axioms, we arrive at the generalnotion of an associator that appeared in Drinfeld’ papers [Dr1, Dr2] in re-lation to his study of the possible generalizations of associativity in algebras.The axioms provide space for a large collection of associators belonging tothe completed algebra of chord diagrams on three strands. Among the as-sociators, there are some with rational coefficients, which proves rationalityof the Kontsevich integral, too.

12.1. The Knizhnik—Zamolodchikov equation and iterated

integrals

In this section, we give the original Drinfeld’s definition of the associator interms of the solutions of the simplest Knizhnik–Zamolodchikov equation.

The Knizhnik—Zamolodchikov (KZ) equation is the basis of the Wess–Zumino–Witten model of conformal field theory [KnZa]. The theory ofKZ type equations has been intensively developed from various approaches:mathematical physics, representation theory and topology [EFK, Var, Kas,Koh4, Oht1]. Our exposition aims at topology and is close to that of thelast three books.

12.1.1. General theory. The notion of the KZ equation can be given inthe following general setting. Let H =

⋃pj=1Hj be a collection of affine

253

254 12. The Drinfeld associator

hyperplanes in the space Cn. Each hyperplane Hj is defined by a (non-homogeneous) linear equation Lj = 0. Let A be a graded C-algebra whichis supposed to be completed (4.6), connected (A.2.21), associative (but ingeneral non-commutative) and with a unit. Fixing a set cj of first-degreehomogeneous elements of A, one for each hyperplane, we can consider adifferential 1-form ω =

∑pj=1 cjd log(Lj) on the complex variety X = Cn \H

with values in A.

Definition. A KZ equation is an equation of the form

(1) dI = ω · I,where I : X → A is the unknown function.

If x1, ..., xn are complex coordinates on X and Lk = λk0 +∑λkjxj for

each k = 1, ..., n, then Equation (1) takes the form

∂I

∂xk=

n∑

j=1

λkjcjLj

I

which is a system of first order linear partial differential equations imposedon a vector-valued function I of several complex variables, and we are inter-ested in complex-analytic solutions of this equation. The main difficulty isthat the coefficients cj as well as the values of I belong to a non-commutativedomain A.

Exercise. One may be tempted to solve the KZ equation as follows:d log(I) = ω, therefore I = exp

∫ω. Explain why this is wrong.

In order that Equation 1 may possess non-zero solutions, the form ωmust satisfy certain conditions. Indeed, taking the differential of both sidesof 1, we get: 0 = d(ωI). Applying the Leibniz rule, using the fact thatdω = 0 and substituting dI = ωI, we see that the necessary condition forintegrability can be written as

(2) ω ∧ ω = 0

It turns out that this condition is not only necessary, but sufficient: ifit holds, then the KZ equation has a unique local solution I0 for any initialvalue setting I0(x0) = a0 (here x0 ∈ X and a0 ∈ A). This fact is standard indifferential geometry where it is called the integrability of flat connections(see, e.g. [KN]). A direct ad hoc proof can be found in [Oht1], Proposition5.2.

Suppose that the integrability condition (2) holds. What can be saidabout the set of all (global) solutions S of Equation (1)? It is easy to seethat solutions can be multiplied on the right by the elements of the algebraA: if I ∈ S and a ∈ A, then Ia ∈ S. This does not mean, however, thatS is a module over the algebra A: the solutions may be multivalued and

12.1. The Knizhnik—Zamolodchikov equation 255

therefore do not form an additive group. To obtain a genuine A-module,one must consider germs of local solutions at any given point x0 ∈ X. Thismodule S(x0) is free of rank 1; it is generated, e.g., by the germ of a localsolution taking value 1 ∈ A at this point.

12.1.2. Monodromy. The reason why the solutions of Equation (1) aremultivalued is the analytic continuation along closed paths in the base spaceX. Indeed, let I0 be the (single-valued) local solution defined in a neighbor-hood of the point x0 ∈ X and satisfying I0(x0) = a0. Let γ : [0, 1]→ X bea closed loop in X, e.g. a continuous mapping such that γ(0) = γ(1) = x0.Since our equation is linear, the solution I0 can be analytically continuedalong γ without running into singularities and thus lead to another localsolution Iγ , also defined in a neighborhood of x0.

Let Iγ(x0) = aγ . Suppose a0 is an invertible element of A. The fact thatS(x0) ∼= A implies that these two solutions are proportional to each other:Iγ = Ia−1

0 aγ . The coefficient a−10 aγ does not depend on a particular choice

of the invertible element a0 ∈ A and the loop γ within a fixed homotopyclass. So we get a homomorphism π1(X)→ A∗ from the fundamental groupof X into the multiplicative group of the algebra A, called the monodromyrepresentation.

12.1.3. Iterated integrals. Both the analytic continuation of the solu-tions and the monodromy representation can be expressed in terms of the1-form ω. Choose a path γ, not necessarily closed, and consider the com-position I γ. This is a function [0, 1] → A which we denote by the sameletter I; it satisfies an ordinary differential equation

(3)d

dtI(t) = ω(γ(t)) · I(t), I(0) = 1 .

The function I takes values in the completed graded algebra A, so it can beexpanded in an infinite series according to the grading:

I(t) = I0(t) + I1(t) + I2(t) + . . . ,

where each term Im(t) is the homogeneous degree m part of I(t).

The form ω is homogeneous of degree 1 (recall that ω =∑cjωj , where

ωj are C-valued 1-forms and cj ’s are elements of A1). Therefore Equation(3) is equivalent to an infinite system of ordinary differential equations

I ′0(t) = 0, I0(0) = 1,I ′1(t) = ω(t)I0(t), I1(0) = 0,I ′2(t) = ω(t)I1(t), I2(0) = 0,. . . . . . . . . . . .

where ω(t) = γ∗ω is the pull-back of the 1-form to the interval [0, 1].


These equations can be solved iteratively, one by one. The first onegives: I0 = const, so that for a basic solution we can take I0(t) = 1. Then,

I1(t) =

∫ t

0ω(t1). Here t1 is an auxiliary variable that ranges from 0 to t.

Coming to the next equation, we now get:

I2(t) =

∫ t

0ω(t2) · I1(t2) =

∫ t

0ω(t2)

(∫ t2

0ω(t1)

)=

∫

0<t1<t2<t

ω(t2) ∧ ω(t1),

Proceeding in the same way, for an arbitrary m we obtain

Im(t) =

∫

0<t1<t2<···<tm<t

ω(tm) ∧ ω(tm−1) ∧ · · · ∧ ω(t1)

Renumbering the variables, we can write the result as follows:

(4) I(t) = 1 +∞∑

m=1

∫

0<tm<tm−1<···<t1<1

ω(t1) ∧ ω(t2) ∧ · · · ∧ ω(tm)

The value I(1) represents the monodromy of the solution over the loop γ.Each iterated integral Im(1) is a homotopy invariant (of “order m”) of γ.Note the resemblance of these expressions to the Kontsevich integral — we’llcome back to that again later.

12.1.4. The formal KZ equation. The general scheme of the previoussection allows for various specializations. The case of KZ equations relatedto Lie algebras and their representations is the most elaborated one (see,e.g., [KnZa, Oht1, Koh4]). We are especially interested in the followingsituation.

Suppose that, in the notation of the previous section, X = Cn \ Hwhere H is the union of the diagonal hyperplanes zj = zk, 1 ≤ j < k ≤n, and the algebra A is the completed algebra of horizontal tangled chord

diagrams Ah(n) for the tangle T consisting of n vertical strings modulothe tangled four-term relation (see 8.6.2 and 8.6.3). In the pictures, we willalways suppose that the strings are oriented upwards. The multiplication inthe algebra Ah(n) is defined by the vertical concatenation of tangled chorddiagrams (to obtain the product xy, one puts x on top of y). The unit is thediagram without chords. The algebra is graded by the number of chords. Itis generated by the diagrams

ujk =

kj

, 1 6 j, k 6 n,


subject to relations (infinitesimal pure braid relations, first appeared in[Koh2])

[ujk, ujl + ukl] = 0, if j, k, l are different,

[ujk, ulm] = 0, if j, k, l,m are different,

where by definition ujk = ukj .

Consider an Ah(n)-valued 1-form ω =1

2πi

∑

16j<k6n

ujkdzj − dzkzj − zk

and the

corresponding KZ equation

(5) dI =1

2πi

( ∑

16j<k6n

ujkdzj − dzkzj − zk

)· I .

This specialization of Equation 1 is referred to as the formal KZ equation.

The integrability condition (2) for the formal KZ equation is the follow-ing identity for a 2-form on X with values in the algebra Ah(n):

ω ∧ ω =∑

16j<k6n16l<m6n

ujkulmdzj − dzkzj − zk

∧ dzl − dzmzl − zm

= 0 .

This identity, in a slightly different notation, was actually proved in Section8.6.9 when we checked the horizontal invariance of the Kontsevich integral.

The space X = Cn \ H is the configuration space of n different (anddistinguishable) points in C. A loop γ in this space may be identified witha pure braid (that is a braid that does not permute the endpoints of thestrings), and the iterated integral formula (4) yields

I(1) =∞∑

m=0

1

(2πi)m

∫

0<tm<···<t1<1

∑

P=(zj ,z′j)DP

m∧

j=1


,

where P (a pairing) is a choice of m pairs of points on the braid, withj-th pair lying on the level t = tj , and DP is the product of m T -chorddiagrams of type ujj′ corresponding to the pairing P . We can see that themonodromy of the KZ equation over γ coincides with the Kontsevich integralof the corresponding braid (see 8.6.5).

12.1.5. The case n = 2. There are some simplifications in the treatment

of Equation 5 for small values of n. In the case n = 2 the algebra Ah(2) isfree commutative with one generator and everything is very simple, as thefollowing exercise shows.

Exercise. Solve explicitly Equation 5 and find the monodromy repre-sentation in the case n = 2.


12.1.6. The case n = 3. The formal KZ equation for n = 3 has the form

dI =1

2πi

(u12d log(z2 − z1) + u13d log(z3 − z1) + u23d log(z3 − z2)

)· I ,

which is a partial differential equation in 3 variables. It turns out that itcan be reduced to an ordinary differential equation.

Indeed, make the substitution

I = (z3 − z1)u

2πi ·G ,

where u := u12 + u13 + u23 and we understand the multiplier as a power

series in the algebra Ah(3):

(z3 − z1)u

2πi = exp

(log(z3 − z1)

2πiu

)

= 1 +log(z3 − z1)

2πiu+

1

2!

log2(z3 − z1)(2πi)2

u2 +1

3!

log3(z3 − z1)(2πi)3

u3 + . . .

The element u and therefore the values of the function (z3− z1)u

2πi com-

mute with all elements of the algebra Ah(3) because of the following lemma.

Lemma. The element u = u12+u13+u23 belongs to the center of the algebra

Ah(3).

Proof. The algebra Ah(3) is generated by the elements u12, u13, and u23,therefore it is enough to show that each of these three elements commuteswith their sum. But the tangled four-term relations mean precisely that

[u12, u13 + u23] = 0, [u13, u12 + u23] = 0, [u23, u12 + u13] = 0 .

The algebra Ah(3) can thus be considered as a free algebra of mixedtype, with two noncommutative generators u12, u23 and one commutativegenerator u.

After the substitution and simplifications the differential equation for Gbecomes

dG =1

2πi

(u12d log

(z2 − z1z3 − z1

)+ u23d log

(1− z2 − z1

z3 − z1

))G .

Denoting z2−z1z3−z1 simply by z, we see that the function G depends only

on z and as such satisfies the following ordinary differential equation (thereduced KZ equation)

(6)dG

dz=

(A

z+

B

z − 1

)G


where A := u122πi , B := u23

2πi . Initially, G was taking values in the algebra

Ah(3) with three generators A, B, u. However, the space of local solutions

of this equation is a free module over Ah(3) of rank 1, so the knowledgeof just one solution is enough. Since the coefficients of Equation 6 do notinvolve u, the equation does have a solution with values in the ring of formalpower series C〈〈A,B〉〉 in two non-commuting variables A and B.

12.1.7. The reduced KZ equation. The reduced KZ equation (6) is aparticular case of the general KZ equation (1), defined by the data n = 1,X = C \ 0, 1, A = C〈〈A,B〉〉, c1 = A, c2 = B.

Equation (6), although it is a first order ordinary differential equation,is not easy to solve, because the solutions take values in a non-commutativeinfinite-dimensional algebra. Two approaches at solving this equation im-mediately come to mind. In the following exercises we invite the reader totry them.

12.1.8. Exercise. Try to find the general solution of Equation (6) by repre-senting it as a series G = G0 +G1A+G2B+G11A

2 +G12AB+G21BA+ . . . ,where the G’s with subscripts are complex-valued functions of z.

12.1.9. Exercise. Try to find the general solution of Equation (6) in theform of a Taylor series G =

∑kGk(z − 1

2)k, where the Gk’s are elements ofthe algebra C〈〈A,B〉〉. (Note that it is not possible to expand the solutionsat z = 0 or z = 1, because they have essential singularities at these points.)

These exercises show that direct approaches do not give much insightinto the nature of solutions of the KZ equation (6). However, one goodthing about this equation is that any solution can be obtained from onebasic solution via multiplication by an element of the algebra C〈〈A,B〉〉.The Drinfeld associator appears as a coefficient between two remarkablesolutions.

Definition. The (Knizhnik-Zamolodchikov) Drinfeld associator ΦKZ is theratio ΦKZ = G−1

1 (z) ·G0(z) of two special solutions G0(z) and G1(z) of thisequation described in the following Lemma.

12.1.10. Lemma. ([Dr1, Dr2]) There exist unique solutions G0(z) andG1(z) of equation (6), analytic in the domain z ∈ C | |z| < 1, |z − 1| < 1and having the following asymptotic behavior:

G0(z) ∼ zA as z → 0 and G1(z) ∼ (1− z)B as z → 1 ,

which means that

G0(z) = f(z) · zA and G1(z) = g(1− z) · (1− z)B ,

where f(z) and g(z) are analytic functions in a neighborhood of 0 ∈ Cwith values in C〈〈A,B〉〉 such that f(0) = g(0) = 1, and the (multivalued)


exponential functions are understood as formal power series, e.g. zA =exp(A log z) =

∑k≥0(A log z)k/k!

Remark. It is often said that the element ΦKZ represents the monodromyof the KZ equation over the horizontal interval from 0 to 1. This phrasehas the following meaning. In general, the monodromy along a path γconnecting two points p and q, is the value at q of the solution, analyticalover γ and taking value 1 at p. If fp and fq are two solutions analyticalover γ with initial values fp(p) = fq(q) = 1, then the monodromy is theelement f−1

q fp. The reduced KZ equation has no analytic solutions at thepoints p = 0 and q = 1, so the general definition of the monodromy cannotbe by applied directly in this case. What we do is we choose some naturalbasic solutions with reasonable asymptotics at these points and define themonodromy as their ratio in the appropriate order.

Proof. Plugging the expression G0(z) = f(z) · zA into Equation (6) we get

f ′(z) · zA + f · Az· zA =

(A

z+

B

z − 1

)· f · zA ,

hence f(z) satisfies the differential equation

f ′ − 1

z[A, f ] =

−B1− z · f .

Let us look for a formal power series solution f = 1 +∑∞

k=1 fkzk with

coefficients fk ∈ C〈〈A,B〉〉. We have the following recurrence equation forthe coefficient of zk−1:

kfk − [A, fk] = (k − adA)(fk) = −B(1 + f1 + f2 + · · ·+ fk−1) ,

where adA denotes the operator x 7→ [A, x]. The operator k− adA is invert-ible:

(k − adA)−1 =∞∑

s=0

adsAks+1

(the sum is well-defined because the operator adA increases the grading), sothe recurrence can be solved

fk =∞∑

s=0

adsAks+1

(−B(1 + f1 + f2 + · · ·+ fk−1)

).

Therefore the desired solution does exist among formal power series. Sincethe point 0 is a regular singular point of our equation (6), it follows (see, e.g.[Wal]) that this power series converges for |z| < 1. We thus get an analyticsolution f(z).

To prove the existence of the second solution, G1(z), it is best to makethe change of undependent variable z 7→ 1 − z which transforms Equation(6) into a similar equation with A and B swapped.


Remark. If the variables A and B were commutative, then the functionexplicitly given as the product zA(1−z)B would be a solution of Equation 6satisfying both asymptotic conditions of Lemma 12.1.10 at once. Therefore,the image of ΦKZ under the abelianization map C〈〈A,B〉〉 → C[[A,B]] isequal to 1.

The next lemma gives another expression for the associator in terms ofthe solutions of equation (6).

12.1.11. Lemma. ([LM2]). Suppose that ε ∈ C, |ε| < 1, |ε − 1| < 1. LetGε(z) be the unique solution of equation (6) satisfying the initial conditionGε(ε) = 1. Then

ΦKZ = limε→0

ε−B ·Gε(1− ε) · εA .

Proof. We rely on, and use the notation of, Lemma 12.1.10. The solutionGε is proportional to the distinguished solution G0:

Gε(z) = G0(z)G0(ε)−1 = G0(z) · ε−Af(ε)−1 = G1(z) · ΦKZ · ε−Af(ε)−1

(the function f , as well as g mentioned below, was defined in Lemma12.1.10). In particular,

Gε(1− ε) = G1(1− ε) · ΦKZ · ε−Af(ε)−1 = g(ε)εB · ΦKZ · ε−Af(ε)−1 .

We must compute the limit

limε→0

ε−Bg(ε)εB · ΦKZ · ε−Af(ε)−1εA ,

which obviously equals ΦKZ because f(0) = g(0) = 1 and f(z) and g(z) areanalytic functions in a neighborhood of zero. The lemma is proved.

12.1.12. The Drinfeld associator and the Kontsevich integral. Putting

ω(z) = Adz

z+B

d(1− z)1− z .

we can rewrite equation (6) as dG = ω · G. This is a particular case ofequation (4) in Section 12.1.1, therefore, its solution Gε(t) can be writtenas a series of iterated integrals

Gε(t) = 1 +

∞∑

m=1

∫

ε<tm<···<t2<t1<t

ω(t1) ∧ ω(t2) ∧ · · · ∧ ω(tm) .

The lower limit in the integrals is ε because of the initial conditionGε(ε) = 1.

We are interested in the value of this solution at t = 1− ε:

Gε(1− ε) = 1 +∞∑

m=1

∫

ε<tm<···<t2<t1<1−ε

ω(t1) ∧ ω(t2) ∧ . . . ω(tm) .


We claim that this series literally coincides with the Kontsevich integral ofthe following tangle

ATε =

0

ε

ε 1−ε 1

1−ε

1

t

z

under the identification A = 12πi , B = 1

2πi . Indeed, on every level tjthe differential form ω(tj) consists of two summands. The first summand

Adtjtj

corresponds to the choice of a pair P = (0, tj) on the first and the

second strings and is related to the chord diagram A = . The second

summand Bd(1−tj)1−tj corresponds to the choice of a pair P = (tj , 1) on the

second and third strings and is related to the chord diagram B = . Thepairing of the first and the third strings does not contribute to the Kontsevichintegral, because these strings are parallel and the correspoding differentialvanishes. We have thus proved the following proposition.

Proposition. The value of the solution Gε at 1−ε is equal to the Kontsevichintegral Gε(1−ε) = Z(ATε). Consequently, the KZ associator coincides withthe regularization of the Kontsevich integral of the tangle ATε:

ΦKZ = limε→0

ε−B · Z(ATε) · εA,

where A = 12πi and B = 1

2πi .

12.2. Calculation of the KZ Drinfeld associator

In this section, following [LM2], we deduce an explicit formula for the Drin-feld associator ΦKZ . It turns out that all coefficients in the correspondingexpansion over the monomials in A and B are values of multiple zeta func-tions (see Section 12.2.11) divided by powers of 2πi.

12.2.1. Put ω0(z) = dzz and ω1(z) = d(1−z)

1−z . Then the 1-form Ω stud-

ied in 12.1.12 is the linear combination ω(z) = Aω0(z) + Bω1(z), where

A = 2πi and B = 2πi . By definition the terms of the Kontsevich integralZ(ATε) represent the monomials corresponding to all choices of one of thetwo summands of ω(tj) for every level tj . The coefficients of these mono-mials are integrals over the simplex ε < tm < · · · < t2 < t1 < 1 − ε of all

12.2. Calculation of the KZ Drinfeld associator 263

possible products of the forms ω0 and ω1. The coefficient of the monomialBi1Aj1 . . . BilAjl is∫

ε<tm<···<t2<t1<1−ε

ω1(t1) ∧ · · · ∧ ω1(ti1)︸︷︷︸i1

∧ω0(ti1+1) ∧ · · · ∧ ω0(ti1+j1)︸︷︷︸j1

∧ . . .

∧ω0(ti1+···+il+1) ∧ · · · ∧ ω0(ti1+···+jl)︸︷︷︸jl

,

where m = i1 +j1 + · · ·+ il+jl. For example, the coefficient of AB2A equals∫

ε<t4<t3<t2<t1<1−ε

ω0(t1) ∧ ω1(t2) ∧ ω1(t3) ∧ ω0(t4) .

We are going to divide the sum of all monomials into two parts, conver-gent Zconv and divergent Zdiv, depending on the behavior of the coefficientsas ε→ 0. We will have Z(ATε) = Zconv + Zdiv and

(1) Φ = limε→0

ε−B · Zconv · ε−A + limε→0

ε−B · Zdiv · ε−A .

Then we will prove that the second limit equals zero and find an explicitexpression for the first one in terms of multiple zeta values. We will see thatalthough the sum Zconv does not contain any divergent monomials, the firstlimit in (1) does.

We pass to exact definitions.

12.2.2. Definition. A non unit monomial in letters A and B is calledconvergent if it starts with an A and ends with a B. Otherwise the monomialis called divergent . We regard the unit monomial 1 as convergent.

12.2.3. Example. The integral∫

a<tp<···<t2<t1<b

ω1(t1) ∧ · · · ∧ ω1(tp) =1

p!logp

( 1− b1− a

)

diverges as b→ 1. It is the coefficient of the monomial Bq in Gε(1−ε) whena = ε, b = 1− ε, and this is the reason to call monomials that start with aB divergent.

Similarly, the integral∫

a<tq<···<t2<t1<b

ω0(t1) ∧ · · · ∧ ω0(tq) =1

q!logq

( ba

)

diverges as a→ 0. It is the coefficient of the monomial Ap in Gε(1−ε) whena = ε, b = 1− ε, and this is the reason to call monomials that end with aaA divergent.


Now consider the general case: integral of a product that contains bothω0 and ω1. For δj = 0 or 1 and 0 < a < b < 1, introduce the notation

Ia,bδ1...δm =

∫

a<tm<···<t2<t1<b

ωδ1(t1) ∧ · · · ∧ ωδm(tm) .

12.2.4. Lemma.

(i) If δ1 = 0 the integral Ia,bδ1...δm converges to a non-zero constant as

b→ 1, and it grows as a power of log(1− b) if δ1 = 1.

(ii) If δm = 1 the integral Ia,bδ1...δm converges to a non-zero constant asa→ 0, and it grows as a power of log a if δm = 0.

Proof. Induction on the number of chords m. If m = 1 then the integralcan be calculated explicitly like in the previous example, and the lemmafollows from the result. Now suppose that the lemma is proved for m − 1chords. By the Fubini theorem the integral can be represented as

Ia,b1δ2...δm=

∫

a<t<b

Ia,tδ2...δm ·dt

t− 1, Ia,b0δ2...δm

=

∫

a<t<b

Ia,tδ2...δm ·dt

t,

for the cases δ1 = 1 and δ1 = 0 respectively. By the induction assumption

0 < c <∣∣∣Ia,tδ2...δm

∣∣∣ <∣∣logk(1− t)

∣∣ for some constants c and k. The comparison

test implies that the integral Ia,b0δ2...δmconverges as b → 1 because Ia,tδ2...δm

grows slower than any power of (1 − t). Moreover,∣∣∣Ia,b0δ2...δm

∣∣∣ > c∫ 1adtt =

−c log(a) > 0 because 0 < a < b < 1.

In the case δ1 = 1 we have

c log(1−b) = c∫ b0dtt−1 <

∣∣∣Ia,b1δ2...δm

∣∣∣ <∣∣∣∫ b0 logk(1− t)d(log(1− t))

∣∣∣ =∣∣∣ log

k+1(1−b)k+1

∣∣∣ ,which proves assertion (i). The proof of assertion (ii) is analogous.

12.2.5. Here is a plan of our subsequent actions.

Let Aconv (resp. Adiv) be the subspace of A = C〈〈A,B〉〉 spanned by allconvergent (resp. divergent) monomials. We are going to define a special

linear map f : A → A which kills divergent monomials and preserves theassociator Φ. Applying f to both parts of equation (1) we will have

(2) Φ = f(Φ) = f(limε→0

ε−B · Zconv · εA)

= f(limε→0

Zconv).

The last equality here follows from the fact that only the unit terms of ε−B

and εA are convergent and therefore survive under the action of f .


The convergent improper integral

(3) limε→0

Zconv = 1 +

∞∑

m=2

∑

δ2,...,δm−1=0,1

I0,10δ2...δm−11 ·ACδ2 . . . Cδm−1B

can be computed explicitly (here Cj = A if δj = 0 and Cj = B if δj = 1).Combining equations (2) and (3) we get

(4) Φ = 1 +∞∑

m=2

∑

δ2,...,δm−1=0,1

I0,11δ2...δm−10 · f(ACδ2 . . . Cδm−1B)

The knowledge of how f acts on the monomials from A leads to the desiredformula for the associator.

12.2.6. Definition of the linear map f : A → A. Consider an algebra

A[α, β] of polynomials in two commuting variables α and β with coeffi-

cients in A. Every monomial in A[α, β] can be written uniquely as βpMαq,

where M is a monomial in A. Define a C-linear map j : A[α, β] → A by

j(βpMαq) = BpMAq. Now for any element Γ(A,B) ∈ A let

f(Γ(A,B)) = j(Γ(A− α,B − β)) .

12.2.7. Lemma. If M is a divergent monomial in A, then f(M) = 0.

Proof. Consider the case when M starts with B, say M = BC2 . . . Cm,where Cj is either A or B. Then

f(M) = j((B − β)M2) = j(BM2)− j(βM2) ,

where M2 = (C2 − γ2) . . . (Cm − γm) with γj = α or γj = β depending onCj . But j(BM2) equals j(βM2) by the definition of j above. The case whenM ends with a A can be done similarly.

12.2.8. One may notice that for any monomial M ∈ A we have f(M) =M +(sum of divergent monomials). Therefore, by the lemma, f is an idem-

potent map, f2 = f , i.e. f is a projection along Adiv (but not onto Aconv).12.2.9. Proposition. f(Φ) = Φ.

Proof. We use the definition of the associator Φ as the KZ Drinfeld associ-ator from Sec. 12.1.7 (see Proposition in Section 12.1.12).

It is the ratio Φ(A,B) = G−11 · G0 of two solutions of the differential

equation (6) from Sec. 12.1.7

G′ =

(A

z+

B

z − 1

)·G


with the asymptotics

G0(z) ∼ zA as z → 0 and G1(z) ∼ (1− z)B as z → 1 .

Consider the differential equation

H ′ =

(A− αz

+B − βz − 1

)·H .

A direct substitution shows that the functions

H0(z) = z−α(1− z)−β ·G0(z) and H1(z) = z−α(1− z)−β ·G1(z)

are its solutions with the asymptotics

H0(z) ∼ zA−α as z → 0 and H1(z) ∼ (1− z)B−β as z → 1 .

Hence we have

Φ(A− α,B − β) = H−11 ·H0 = G−1

1 ·G0 = Φ(A,B) .

Therefore

f(Φ(A,B)) = j(Φ(A− α,B − β)) = j(Φ(A,B)) = Φ(A,B)

because j acts as the identity map on the subspace A ⊂ A[α, β]. Theproposition is proved.

12.2.10. To compute Φ according to formula (4) we must find the inte-

grals I0,10δ2...δm−11 and the action of f on the monomials. Let us compute

f(ACδ2 . . . Cδm−1B) first.

Represent the monomial M = ACδ2 . . . Cδm−1B in the form

M = Ap1Bq1 . . . AplBql

for some positive integers p1, q1, . . . , pl, ql. Then

f(M) = j((A− α)p1(B − β)q1 . . . (A− α)pl(B − β)ql) .

We are going to expand the product, collect all β’s on the left and all α’s onthe right, and then replace β by B and α by A. To this end let us introducethe following multi-index notations:

r = (r1, . . . , rl); i = (i1, . . . , il); s = (s1, . . . , sl); j = (j1, . . . , jl);

p = r + i = (r1 + i1, . . . , rl + il); q = s + j = (s1 + j1, . . . , sl + jl);

|r| = r1 + · · ·+ rl; |s| = s1 + · · ·+ sl;

(p

r

)=

(p1

r1

)(p2

r2

). . .

(plrl

);

(q

s

)=

(q1s1

)(q2s2

). . .

(qlsl

);

(A,B)(i,j) = Ai1 ·Bj1 · · · · ·Ail ·Bjl


We have

(A− α)p1(B − β)q1 . . . (A− α)pl(B − β)ql =

∑

06r6p06s6q

(−1)|r|+|s|(p

r

)(q

s

)· β|s|(A,B)(i,j)α|r| ,

where the inequalities 0 6 r 6 p and 0 6 s 6 q mean 0 6 r1 6 p1, . . . ,0 6 rl 6 pl, and 0 6 s1 6 q1,. . . , 0 6 sl 6 ql. Therefore

(5) f(M) =∑

06r6p06s6q

(−1)|r|+|s|(p

r

)(q

s

)·B|s|(A,B)(i,j)A|r| .

12.2.11. To complete the formula for the associator we need to computethe coefficient I0,1

1δ2...δm−10 of f(M). It turns out that, up to a sign, they are

equal to some values of the multivariate ζ-function

ζ(a1, . . . , an) =∑

0<k1<k2<···<kn

k−a11 . . . k−an

n

where a1, ..., an are positive integers (see [LM1]). Namely, the coefficientsin question are equal, up to a sign, to the values of ζ at integer points(a1, . . . , an) ∈ Zn, which are called (multiple zeta values, or MZV for short).Multiple zeta values for n = 2 were first studied by L. Euler in 1775. Hispaper [Eu] contains several dozens interesting relations between MZVs andvalues of the univariate Riemann’s zeta function. Later, this subject wasalmost forgotten for more than 200 years until D. Zagier and M. Hoffmanrevived a general interest to MZVs by their papers [Zag3], [Hoff].

Exercise. The sum converges if and only if an > 2.

12.2.12. Remark. Different conventions about the order of arguments inζ.

12.2.13. Proposition. For p > 0 and q > 0 let

(6) η(p,q) := ζ(1, . . . , 1︸︷︷︸ql−1

, pl + 1, 1, . . . , 1︸︷︷︸ql−1−1

, pl−1 + 1, . . . 1, . . . , 1︸︷︷︸q1−1

, p1 + 1) .

Then

(7) I0,10...0︸︷︷︸p1

1...1︸︷︷︸q1

...... 0...0︸︷︷︸pl

1...1︸︷︷︸ql

= (−1)|q|η(p,q) .


The calculations needed to prove the proposition, are best organised interms of the (univariate) multiple polylogarithm1 function defined by theseries

(8) Lia1,...,an(z) =∑

0<k1<k2<···<kn

zkn

ka11 . . . kan

n,

which obviously converges for |z| < 1.

12.2.14. Lemma. For |z| < 1

Lia1,...,an(z) =

∫ z

0Lia1,...,an−1(t)

dt

t, if an > 1

−∫ z

0Lia1,...,an−1(t)

d(1− t)1− t , if an = 1 .

Proof. The lemma follows from the next identities whose proofs we leaveto the reader as an exercise.

ddzLia1,...,an(z) =

1z · Lia1,...,an−1(z) , if an > 1

11−z · Lia1,...,an−1(z) , if an = 1 ;

ddzLi1(z) = 1

1−z .

12.2.15. Proof of proposition 12.2.13. From the previous lemma wehave

Li1,1,...,1︸︷︷︸ql−1

, pl+1, 1,1,...,1︸︷︷︸ql−1−1

, pl−1+1, ..., 1,1,...,1︸︷︷︸q1−1

, p1+1(z) =

= (−1)q1+···+ql∫

0<tm<···<t2<t1<z

ω0(t1) ∧ · · · ∧ ω0(tp1)︸︷︷︸p1

∧

∧ω1(tp1+1) ∧ · · · ∧ ω1(tp1+q1)︸︷︷︸q1

∧ · · · ∧ ω1(tp1+···+pl+1) ∧ · · · ∧ ω1(tp1+···+ql)︸︷︷︸ql

=

= (−1)|q|I0,z0...0︸︷︷︸p1

1...1︸︷︷︸q1

...... 0...0︸︷︷︸pl

1...1︸︷︷︸ql

.

1It is a generalization of Euler’s dilogarithm Li2(z) we used on p.216, and a special case of

multivariate polylogarithm

Lia1,...,an(z1, . . . , zn) =

X0<k1<k2<···<kn

zk1

1 . . . zkn

n

ka1

1 . . . kan

n

introduced by A. Goncharov in [Gon].


Note that the multiple polylogarithm series (8) converges for z = 1 in thecase an > 1. This implies that if pl > 1 (which holds for a convergentmonomial) we have

η(p,q) = ζ(1, . . . , 1︸︷︷︸ql−1

, pl + 1, 1, . . . , 1︸︷︷︸ql−1−1

, pl−1 + 1, . . . 1, . . . , 1︸︷︷︸q1−1

, p1 + 1)

= Li1,1,...,1︸︷︷︸ql−1

, pl+1, 1,1,...,1︸︷︷︸ql−1−1

, pl−1+1, ..., 1,1,...,1︸︷︷︸q1−1

, p1+1(1)

= (−1)|q|I0,10...0︸︷︷︸p1

1...1︸︷︷︸q1

...... 0...0︸︷︷︸pl

1...1︸︷︷︸ql

.

The Proposition is proved.

12.2.16. Explicit formula for the associator. Combining equations(4), (5), and (7) we get the following formula for the associator.

Φ = 1+∞∑

m=2

∑

0<p,0<q|p|+|q|=m

η(p,q) ·∑

06r6p06s6q

(−1)|r|+|j|(p

r

)(q

s

)·B|s|(A,B)(i,j)A|r|

where i and j are multi-indices of the same length, p = r + i, q = s + j, andη(p,q) is the multiple zeta value given by (6).

This formula was obtained by Le and Murakami in [LM1, LM2, LM4].

12.2.17. Example. Degree 2 terms of the associator. There is onlyone possibility to represent m = 2 as the sum of two positive integers:2 = 1 + 1. So we have only one possibility for p and q: p = (1), q = (1). Inthis case η(p,q) = ζ(2) = π2/6 according to (6). The multi-indices r ands have length 1 and thus consist of a single number r = (r1) and s = (s1).There are two possibilities for each of them: r1 = 0 or r1 = 1, and s1 = 0

or s1 = 1. In all these cases the binomial coefficients

(p

r

)and

(q

s

)equal

1. We arrange all the possibilities in the following table.

r1 s1 i1 j1 (−1)|r|+|j| ·B|s|(A,B)(i,j)A|r|

0 0 1 1 −A B

0 1 1 0 B A

1 0 0 1 B A

1 1 0 0 −B A


Hence, for the degree 2 terms of the associator we get the formula:

−ζ(2)[A,B] = − ζ(2)

(2πi)2[a, b] =

1

24[a, b] ,

where a = (2πi)A = , and b = (2πi)B = .

12.2.18. Example. Degree 3 terms of the associator. There are twoways to represent m = 3 as the sum of two positive integers: 3 = 2 + 1and 3 = 1 + 2. In each case either p = (1) or q = (1). So l = 1 and bothmulti-indices consist of just one number p = (p1), q = (q1). Therefore allother multi-indices r, s, i, j also consist of one number.

Here is the corresponding table.

p1 q1 η(p,q) r1 s1 i1 j1 (−1)|r|+|j|(p

r

)(q

s

)·B|s|(A,B)(i,j)A|r|

0 0 2 1 −A A B

0 1 2 0 B A A

1 0 1 1 2A B A2 1 ζ(3)

1 1 1 0 −2B A A

2 0 0 1 −B A A

2 1 0 0 B A A

0 0 1 2 A B B

1 0 0 2 −B B A

0 1 1 1 −2B A B1 2 ζ(1, 2)

1 1 0 1 2B B A

0 2 1 0 B B A

1 2 0 0 −B B A

Using the Euler identity ζ(1, 2) = ζ(3) (see section 12.2.20) we can sum upthe degree 3 part of Φ into the formula

ζ(3)(−A A B + 2A B A−B A A+A B B − 2B A B +B B A

)

= ζ(3)(−[A,[A,B

]]−[B,[A,B

]])= − ζ(3)

(2πi)3[a+ b, [a, b]

].

12.2.19. Example. Degree 4 terms of the associator. Proceeding inthe same way and using the the identities from Sec.12.2.20:

ζ(1, 1, 2) = ζ(4) = π4/90, ζ(1, 3) = π4/360, ζ(2, 2) = π4/120 ,


we can write out the associator Φ up to degree 4:

ΦKZ = 1 +1

24[a, b] − ζ(3)

(2πi)3[a+ b, [a, b]

]− 1

1440

[a,[a, [a, b]

]]

− 1

5760

[a,[b, [a, b]

]]− 1

1440

[b,[b, [a, b]

]]+

1

1152[a, b]2

+ (terms of order > 4) .

12.2.20. Multiple zeta values. There are a lot of relations between MZV’s

and powersof π. Some of them, like ζ(2) = π2

6 or ζ(1, 2) = ζ(3), were al-ready known to Euler. The last one can be obtained in the following way.According to (6) and (7) we have

ζ(1, 2) = η((1), (2)) = I0,1011 =

∫

0<t3<t2<t1<1

ω0(t1) ∧ ω1(t2) ∧ ω1(t3)

=

∫

0<t3<t2<t1<1

dt1t1∧ d(1− t2)

1− t2∧ d(1− t3)

1− t3.

The change of variables (t1, t2, t3) 7→ (1 − t3, 1 − t2, 1 − t1) transforms thelast integral to

∫

0<t3<t2<t1<1

d(1− t3)1− t3

∧ dt2t2∧ dt1t1

= −∫

0<t3<t2<t1<1

ω0(t1) ∧ ω0(t2) ∧ ω1(t3) = −I0,1001 = η((2), (1)) = ζ(3) .

In the general case a similar change of variables

(t1, t2, . . . , tm) 7→ (1− tm, . . . , 1− t2, 1− t1)gives the identity

I0,10...0︸︷︷︸p1

1...1︸︷︷︸q1

...... 0...0︸︷︷︸pl

1...1︸︷︷︸ql

= (−1)mI0,10...0︸︷︷︸

ql

1...1︸︷︷︸pl

...... 0...0︸︷︷︸ql

1...1︸︷︷︸pl

.

By (7), we have

I0,10...0︸︷︷︸p1

1...1︸︷︷︸q1

...... 0...0︸︷︷︸pl

1...1︸︷︷︸ql

= (−1)|q|η(p,q),

I0,10...0︸︷︷︸

ql

1...1︸︷︷︸pl

...... 0...0︸︷︷︸ql

1...1︸︷︷︸pl

= (−1)|p|η(q,p),

where the bar denotes the inversion of a sequence: p = (pl, pl−1, . . . , p1),q = (ql, ql−1, . . . , q1).


Since |p|+ |q| = m, we deduce that

η(p,q) = η(q,p),

This relation is called the duality relation between MZV’s. After the con-version from η to ζ according to equation (6), the duality relations becomepicturesque and unexpected.

2B AB ABA2

Figure 12.1. Rotation of a tangle through 180

—!! write about the rotation of the picture by 180 and its relation toduality—

As an example, we give a table of all nontrivial duality relations of weightm ≤ 5:

p q q p relation

(1) (2) (2) (1) ζ(1, 2) = ζ(3)

(1) (3) (3) (1) ζ(1, 1, 2) = ζ(4)

(1) (4) (4) (1) ζ(1, 1, 1, 2) = ζ(5)

(2) (3) (3) (2) ζ(1, 1, 3) = ζ(1, 4)

(1, 1) (1, 2) (2, 1) (1, 1) ζ(1, 2, 2) = ζ(2, 3)

(1, 1) (2, 1) (1, 2) (1, 1) ζ(2, 1, 2) = ζ(3, 2)

The reader may want to check this table by way of exercise.

There are other relations between the multiple zeta values that do notfollow from the duality law. Let us quote just a few.

1. Euler’s relations:

ζ(1, n− 1) + ζ(2, n− 2) + · · ·+ ζ(n− 2, 2) = ζ(n),(9)

ζ(m) · ζ(n) = ζ(m,n) + ζ(n,m) + ζ(m+ n) .(10)

2. Relations obtained by Le and Murakami [LM1] computing the Kont-sevich integral of the unknot by the combinatorial procedure explained below


in Section 12.3 (the first one was earlier proved by M. Hoffman [Hoff]):

ζ(2, 2, . . . , 2︸︷︷︸m

) =π2m

(2m+ 1)!(11)

( 1

22n−2− 1)ζ(2n)− ζ(1, 2n− 1) + ζ(1, 1, 2n− 1)− . . .(12)

+ζ(1, 1, . . . , 1︸︷︷︸2n−2

, 2) = 0 .

For multiple zeta values of weight 4, these relations imply four indepen-dent identities:

ζ(1, 3) + ζ(2, 2) = ζ(4),

ζ(2)2 = 2ζ(2, 2) + ζ(4),

ζ(2, 2) =π4

120,

−3

4ζ(4)− ζ(1, 3) + ζ(1, 1, 2) = 0.

Solving these equations one by one and using the identity ζ(2) = π2/6,we find the values of all MZVs of weight 4: ζ(2, 2) = π4/120, ζ(1, 3) =π4/360, ζ(1, 1, 2) = ζ(4) = π4/90.

There exists an extensive literature about the relations between MZV’s,e.g. [BBBL, Car2, Hoff, HoOh, OU], and the interested reader is invitedto consult it.

An attempt to overview the whole variety of relations between MZV’swas undertaken by D. Zagier [Zag3]. Call the weight of a multiple zetavalue ζ(n1, . . . , nk) the sum of all its arguments w = n1 + · · ·+ nk. Let Zwbe the vector subspace of the reals R over the rationals Q spanned by allMZV’s of a fixed weight w. For completeness we put Z0 = Q and Z1 = 0.Denote the formal direct sum of all Zw by Z• :=

⊕w≥0Zw.

Proposition. The vector space Z• forms a graded algebra over Q, i.e. Zu ·Zv ⊆ Zu+v.

Euler’s product formula (10) illustrates this statement. A proof may befound in [Gon]. D. Zagier made a conjecture about the Poincare series ofthis algebra.

Conjecture ([Zag3]).∞∑

w=0

dimQ(Zw) · tw =1

1− t2 − t3 .


This series turns out to be related to the dimensions of various subspacesin the primitive space of the chord diagram algebra A (see [Br, Kre]).

12.2.21. Logarithm of the KZ associator modulo the second com-mutant. The associator ΦKZ is group-like (see exercise 3 at the end of thechapter). Therefore its logarithm can be expressed as a Lie series in vari-ables A and B. Let L be the complement of a free Lie algebra generated byA and B and L′′ := [[L,L], [L,L]] its second commutant. We can considerL as a subspace of C〈〈A,B〉〉. V. Drinfeld [Dr2] found the following formula

log ΦKZ =∑

k,l>0

ckl adlB adkA [A,B] (mod L′′) ,

where the coefficients ckl are defined by the generating function

1 +∑

k,l>0

ckluk+1vl+1 = exp

( ∞∑

n=2

ζ(n)

(2πi)nn

(un + vn − (u+ v)n

))

through the univariate zeta function ζ(n) :=∑∞

k=1 k−n. In particular, ckl =

clk and ck0 = c0k = − ζ(k+2)(2πi)k+2 .

12.3. Combinatorial construction of the Kontsevich integral

12.3.1. The multiplicativity property (Proposition 8.6.6) of the Kontsevichintegral suggests an idea to try something similar to what we did for quan-tum invariants in Sections 2.6.5 and 2.6.6, and get a sort of combinatorialdefinition of the integral. Different versions of such a definition were pro-posed in several papers ([BN2, Car1, LM1, LM2, Piu]) and treated inseveral books ([Kas, Oht1]).

The program is to cut up a knot into a number of standard simpletangles, compute the Kontsevich integral for each of them and then recoverthe integral of the whole knot from these simple pieces. To implementthis program it would be nice to represent a knot by the simplest possiblediagram. A closed braid is a good candidate for such a representation. Inthis case we have essentially only one type of simple tangles: crossings, whichare generators of the braid group and their inverses. This approach looksvery promising because for one crossing the computation of the Kontsevichintegral is fairly easy (see the exercise in 8.6.6).

The situation, however, is not as simple as it looks at the first glance.The problem is that in general the Kontsevich integral of a tangle does de-pend on the specific position of its boundary points. It is true (again byExercise 8.6.6) that for a single crossing the Kontsevich integral does notdepend on the boundary points. So we do not have this difficulty here. But

12.3. Combinatorial construction of the Kontsevich integral 275

we have it for the minimum and maximum points. Besides this the Kontse-vich integral of a horizontal slice containing a simple crossing involves ‘long’chords connecting a braiding string with another string far away from thecrossing. The integral of such a slice is different from just the integral com-puted in Exercise 8.6.7. To get rid of such parasitic ‘long’ chords we needsome argument similar to Lemma 8.7.2 which would allow us to shrink thetangle under consideration into a narrow box. Basically we need not justa decomposition of a knot diagram into tangles but rather a family of de-compositions depending on a parameter ε→ 0 so that the typical horizontalsizes in every elementary tangle (a crossing or a minimum/maximum point)are whole powers of ε. In this setting, although the Kontsevich integral ofeach tangle does depend on ε, we can learn much about its asymptotics asε → 0 and thus derive useful information about the Kontsevich integral ofthe whole knot.

The three types of elementary tangles are not enough to represent theknot diagram in the way we wish. To connect the elementary pieces, wemust use one more tangle type that relates different horizontal distanceslike this:

The Kontsevich integral of this tangle is more or less the same thing asthe Drinfeld associator. As soon as we know an explicit expression for theDrinfeld associator, the Kontsevich integral of an arbitrary knot can befound by a purely combinatorial procedure.

We now turn to precise definitions.

12.3.2. Definition. A parenthesized presentation of a knot is a diagram ofthe knot which is portioned into a number of horizontal slices satisfying thefollowing conditions2.

At every boundary level of a slice (dashed lines in the pictures below)the distances between various strings are proportional to different powers ofa small number ε. With such a level we associate a parenthesis structureenclosing a pair of the closest strings (at the distance of the highest powerof ε) in parentheses, then put parentheses around the strings whose mutual

2In fact, we are speaking about a family of such diagrams depending on a parameter ε ∈

(0, ε0), and the expression ‘proportional to’ below means ‘asymptotically equal as ε→ 0’.


distance is the next power of ε, and so on. Here is an example

ε2 ε3

ε4 ε5 ε3 ε

a b c d e f

−−−−−→ (a(bc))(d(ef))

−−−−−→ ((a(bc))d)(ef)

Every slice contains exactly one special event and several strictly verti-cal strings which are farther away (at a lower power of ε) from any stringparticipating in the event than its width. There are three types of specialevents (with all possible choices of the orientations):

— a max/min event: or ;

— a braiding event:a b

ora b

,

where the strings a and b may be replaced by bunches of parallel stringswhich are closer to each other than the width of this event;

— an associativity event: a b c or ab

c ,

where again the strings a, b, and c may be replaced by bunches of parallelstrings such that any string is closer to the strings in its own bunch than toany string from other bunches. In the example above, the string a is replacedby a bunch, a-bunch, of three strings, a, b, c, the b-bunch consists of a singlestring d, and the c-bunch contains two strings e and f . The distance betweena- and b-bunches is about ε3 while the distance between strings inside thea-bunch is at most ε4. The similar is true for b- and c-bunches.

This is the end of the definition.

The following proposition is obvious.

Proposition. For every knot there is a parenthesized presentation.

12.3.3. Example. The diagram of the trefoil knot 31 shown in Fig. 12.2exemplifies the previous definition and its relation to the parenthesis struc-tures and associativity.

Entries in the last column of Figure 12.2 represent the combinatorialKontsevich integrals of slices with a single event. This notation, in particularS3, will be explained in Sections 12.3.5 and 12.3.6.


t9 ∅ mint8 (a b)

id⊗min⊗idt7 ((a (c d)) b)

S3(Φ−1)⊗ id

t6 (((a c) d) b)R−1 ⊗ id⊗ id

t5 (((c a) d) b)R−1 ⊗ id⊗ id

t4 (((a c) d) b)R−1 ⊗ id⊗ id

t3 (((c a) d) b)S3(Φ)⊗ id

t2 ((c (a d)) b)id⊗max⊗id

t1 (c b)max

t0 ∅

a b

c d

c b

∼ε2

∼ε

∼ε2 ∼1

Figure 12.2. Parenthesized presentation of the left trefoil

12.3.4. Exercise. Find a parenthesized presentation of the figure eightknot (41 in the knot table on page 27).

A possible answer is shown in Fig. 12.3.

∼1∼ε2

∼ε∼ε2

∼ε

∼ε2

Figure 12.3. Parenthesized presentation of the figure eight knot

12.3.5. The combinatorial Kontsevich integral. As mentioned before,a parenthesized presentation is actually a family of knot diagrams depending


on a parameter ε. Instead of computing the Kontsevich integral for eachspecific tangle we can compute only its asymptotics as ε approaches zero.This is an easier task for two reasons. First, the asymptotics does notdepend on the concrete positions of the endpoints of the tangle. Secondly,according to Lemma 8.7.2 we can disregard the ‘long’ chords. In exercisesto Chapter 8 we saw that very often the Kontsevich integral of a tanglediverges as ε→ 0. So we must explain the usage of the word “asymptotic”here. In fact, we mean the limit, as ε→ 0, of a certain regularization of theseries of tangled chord diagrams. Examples of various regularizations andthe corresponding limits are present in problems 13 – 15 of Chapter 8. Weencode the asymptotical behavior of the Kontsevich integral for each of thethree special events as an infinite series of tangled chord diagrams which wecall the combinatorial Kontsevich integral of the event.

Definition. The combinatorial Kontsevich integral Zcomb(K) of a knot Kis the product of the series of tangled chord diagrams corresponding to theevents of a parenthesized presentation of K.

The combinatorial Kontsevich integral of the special events is defined asfollows.

— (max/min). The asymptotics of the Kontsevich integral as ε→ 0 ofsuch a tangle is the corresponding tangled chord diagram without chords.We denoted it by max (min).

— (braiding). There are no ‘long’ chords here. For the two braidingstrings the Kontsevich integral was computed in 8.6.6. We denote it byR. Following [Oht1] we introduce a diagrammatic notation for the variousseries of tangled chord diagrams.

R :=exp( )2

:= · exp(

2

)

= +1

2+

1

2 · 22+

1

3! · 23+ . . . ,

R−1 :=2exp(− )

:= · exp(−

2

)

= − 1

2+

1

2 · 22− 1

3! · 23+ . . . ,

where, as usual in tangled chord diagrams, the sign of a crossing (overcross-ing or undercrossing) is irrelevant. For a generalized braiding tangle, wherewe have two braiding bunches of strings, the combinatorial Kontsevich in-tegral can be expressed in terms of R using the operation ∆ from Section12.3.6.


— (associativity tangle). The situation here is more complicated becausethe Kontsevich integral of the associativity tangle diverges as ε→ 0. Propo-sition 12.1.12 suggests a way to regularize it. The limit of this regularizationis called Φ. Diagrammatically, we denote it as follows.

Φ := Zcomb

( )=: Φ .

For a generalized associativity tangle with many strings the combinatorialKontsevich integral can be expressed in terms of Φ using the operations fromthe next subsection (see also exercises 14 and 15 in Chapter 8).

12.3.6. Operations on tangled chord diagrams. Definition. To takethe tensor product T1 ⊗ T2 of two T -chord diagrams T1 and T2 means toplace one diagram next to the other, side by side:

T1 ⊗ T2 := T1 T2

Definition. The T -chord diagram id is just a single vertical string withno chords on it.

For example, id⊗min⊗id is the tangled chord diagram without chords,having 4 strings, out of which the first and the last one are vertical, whilethe second and the third strings form the elementary event min. It is thetangle between levels t8 and t7 of Figure 12.2 considered as a tangled chorddiagram.

Definition. Suppose the i-th string of a tangled chord diagram T con-tains k endpoints of chords of T . Then we define Si(T ) to be equal to

(−1)k · T , where T is obtained from T by reversing the orientation of thei-th string. Here is an example:

S3

( )= (−1)3 .

The notation ‘S’ for the operation just introduced is chosen because ofits analogy with the antipode in Hopf algebras (see p. 345). There exist thecounterparts of comultiplication and counit, too. We will use them later inSection 12.4.

Definition. The operation ∆i acts on a tangled chord diagram T bydoubling the i-th string of T and taking the sum over all possible lifts of theendpoints of chords of T from the i-th string to one of the two new strings.


For example,

∆1

( )= + + + .

Here we count the strings of T by their bottom points from left to right. Thisoperation can be used to express the combinatorial Kontsevich integral ofa generalized associativity tangle in terms of the combinatorial Kontsevichintegral of a simple associativity tangle:

Zcomb

( )= ∆3

(Zcomb

( ))= ∆3(Φ) .

Definition. The operation εi acts from A(n) to A(n− 1). Its value ona tangled chord diagram T is defined by the following rule. If there is atleast one endpoint on the i-th string of T then εi(T ) = 0, otherwise εi(T ) isobtained from T by removing the i-th string.

Examples. εi( ) = for any i, ε1( ) = ε2( ) = 0, ε3( ) = .

12.3.7. Example of computation. To compute the combinatorial Kont-sevich integral of the trefoil we must multiply out all tangled chord diagramsin the third column of Figure 12.2:

Zcomb(31) =

S

2exp(− )

S3(Φ)

3(Φ−1)

2exp(− )

2exp(− ) =

max·(id⊗max⊗id)·(S3(Φ)⊗ id)·(R−1 ⊗ id⊗ id)·(R−1 ⊗ id⊗ id)·(R−1 ⊗ id⊗ id)·(S3(Φ

−1)⊗ id)·(id⊗min⊗id)·min

That is,

Z(31) = S3(Φ)R−3S3(Φ−1),

where the product from left to right corresponds to the multiplication oftangles from top to bottom. Up to degree 2, we have: Φ = 1+ 1

24 [a, b]+ . . . ,

R = X(1+ 12a+ 1

8a2+. . . ), where X means that the two strands in each term

of the series must be crossed over at the top. The operation S3 changes theorientation of the third strand, which means that S3(a) = a and S3(b) = −b.Therefore, S3(Φ) = 1 − 1

24 [a, b] + . . . , S3(Φ−1) = 1 + 1

24 [a, b] + . . . , R−3 =

X(1− 32a+ 9

8a2+. . . ) and Z(31) = (1− 1

24 [a, b]+. . . )X(1− 32a+ 9

8a2+. . . )(1+

124 [a, b] + . . . ) = 1− 3

2Xa− 124abX + 1

24baX + 124Xab− 1

24Xba+ 98Xa

2 + . . .Closing these diagrams into the circle, we see that in the algebra A we haveXa = 0 (by the framing independence relation), then baX = Xab = 0 (by


the same relation, because these diagrams consist of two parallel chords) and

abX = Xba = Xa2 = . The result is: Z(31) = 1 + 2524 + . . . . The

final Kontsevich integral of the trefoil (in the multiplicative normalization,

see page 211) is thus equal to I ′(31) = Z(31)/Z(H) = (1+ 2524 +. . . )/(1+

124 + . . . ) = 1 + + . . .

12.3.8. Main theorem about the combinatorial Kontsevich inte-gral. Theorem. ([LM3]) The combinatorial Kontsevich integral of a knotis equal to the ordinary Kontsevich integral:

Zcomb(K) = Z(K) .

Here is the idea of a proof.

We take a parenthesized presentation of K. For a fixed small numberε we compute the Kontsevich integrals of every simple tangle participatingin the presentation. Then we insert the factors εA, ε−A (‘buffers’) betweenthe integrals of the simple tangles in order to regularize the Kontsevich inte-grals of associativity tangles and in such a way that totally they cancel eachother and the entire Kontsevich integral of the knot K does not change. Theinsertion of these regularization buffers is the most crucial part of the con-struction. The buffers depend on the parentheses structure on the boundaryof two factors in the tangle decjmposition. Take an example:

εε2

ε3

ε 1

T1

T2

Then the regularization buffers inserted to the bottom of Z(T1) and tothe top of Z(T2) are the following.


Z(T1)

Z(T2)

ε−32πi

ε3

2πi

ε−12πi

S1( )

ε1

2πiS1( )

ε−22πi

∆2( )

ε2

2πi∆2( )

ε−12πi

∆21(S2( ))

ε1

2πi∆2

1(S2( ))

−−−−−→ε→0

Zcomb(T1)

−−−−−→ε→0

Zcomb(T2)

After that we take the limit as ε approaches zero. It remains to prove thatthe regulzrized limits for the elementary tangles are the asymptotic series ofT -chord diagrams denoted above by min, max, R, and Φ. We do not give adetailed proof of this, referring the reader to [LM3]. Also some exercises atthe end of chapter 8 may give the reader the feeling that this is true. In theend the Kontsevich integral Z(K) turns out to be composed of the seriesmin, max, R, and Φ, i.e. equals the combinatorial Kontsevich integral.

One of the main things which ensures success to this program is thecommutativity of multiplication of the horizontal chord diagrams on twostrings. In particular, it implies that

ε−α

2πi ·R · ε α2πi = R

Another property is that the regularization factors, like εα

2πi , diverge veryslowly. Indeed, the coefficient of degree n in the series expansion of sucha factor is proportional to logn(εα), and therefore it grows slower than anynegative power of ε. This property can be used as follows. Consider, forexample, a braiding tangle BTε from page 216

BTε =

1

ε

.

Its Kontsevich integral Z(BTε) contains the terms with ‘long’ chords con-necting a braiding string with the rightmost vertical string. The coefficientof such a diagram is proportional to ε. Hence the conjugate of the Kontse-

vich integral ε−α2πi Z(BTε)ε

α2πi will converge as ε → 0. Indeed the terms

12.4. General associators 283

of Z(BTε) with a ‘long’ chord overweight the divergence of the conjugatingfactors and the limit of such a term would be zero as ε→ 0. For the termsof Z(BTε) without ‘long’ chords we can use the commutativity of multi-plication of T -chord diagrams on two strings, and then for such terms theconjugating factors cancel each other. Therefore, passing to the limit weobtain

limε→0

ε−α2πi Z(BTε)ε

α2πi = R⊗ id .

A similar argument also works for tangles containing a minimum (max-imum) point.

12.3.9. Example. Consider the parenthesized presentation of the trefoil31 from 12.3.3. We consider the Kontsevich integral of every slice betweenthe consecutive levels tj and tj−1 and multiply them with appropriate reg-ularization factors (see Figure 12.4). The whole integral does not changebecause we arrange these additional factors in such a way that they canceleach other pairwise. Moreover, the integral does not depend on ε since it isinvariant under the horizontal deformations. So it remains the same if wepass to the limit as ε→ 0. The limits of the corresponding pieces are shownin the right column. Their tangled product is precisely the combinatorialKontsevich integral.

12.4. General associators

Let A(n) be the algebra of tangled chord diagrams on n strings (we willalways depict them vertical and oriented upwards). (In distinction with thepreviously considered algebra Ah(n), the chords need not be horizontal.)The space A(n) is graded by half the number of chords in the diagram. We

denote the graded completion of this graded space by A(n).

12.4.1. Definition. An associator Φ is an element of the algebra A(3)satisfying the following axioms:

• (strong invertibility) ε1(Φ) = ε2(Φ) = ε3(Φ) = 1 (operations εi aredefined on page 280; in particular, this property means that theseries Φ starts with 1 and thus represents an invertible element of

the algebra A(3)).

• (skew symmetry) Φ−1 = Φ321, where Φ321 is obtained from Φ bychanging the first and the third strings (i.e. by adding a permuta-tion (321) both to the top and the bottom of each diagram appear-ing in the series Φ).

• (pentagon relation) (id⊗ Φ) · (∆2Φ) · (Φ⊗ id) = (∆3Φ) · (∆1Φ).


Diagrammatically this relation can be depicted as follows.

=

a b

c d

((ab)c)d

(a(bc))d

a((bc)d) a(b(cd))

(ab)(cd)

Φ⊗ id

∆2Φ

id⊗ Φ

∆1Φ

∆3Φ

• (hexagon relation) Φ · (∆2R) · Φ = (id⊗R) · Φ · (R⊗ id).

=

a cb

(ab)c

a(bc)

(bc)a

b(ca)

b(ac))

(ba)c

Φ

∆2R

Φ

R⊗ id

Φ

id⊗R

This relation also contains the element R defined as follows

R = · exp(

2

).

A version of the last two relations appears in abstract category the-ory where they make part of the definition of a monoidal category(see [ML, Sec.XI.1]).

12.4.2. Theorem. The Knizhnik–Zamolodchikov Drinfeld associator ΦKZ

satisfies the axioms above.

Proof. The main observation is that the pentagon and the hexagon relationshold because ΦKZ can be expressed through the Kontsevich integral andtherefore possesses the property of horizontal invariance. The details of theproof are as follows.

Property 1 immediately follows from the explicit formula 12.2.16 for theassociator ΦKZ, which shows that the series starts with 1 and every term


appearing with non-zero coefficient has endpoints of chords on each of thethree strings.

Property 2 We must prove that

12.4.3. Lemma.

Φ = Φ−1 .

Indeed, the inverse associator Φ−1 is equal to the combinatorial Kont-sevich integral of the following parenthesized tangle which can be deformedto the picture on the right.

=

R−1 ⊗ id

∆2(R−1)

Φ

R⊗ id

∆2(R)

Therefore, Φ−1 = (R−1 ⊗ id) ·∆2(R−1) · Φ · (R⊗ id) ·∆2(R).

Now we have

(R⊗ id) ·∆2(R) =

exp

(+

2

)

exp

(2

)

=exp

(+

2

)

exp

(2

)


=

exp

(+ +

2

)

=

exp

(+ +

2

)

Similarly we can get

∆2(R−1) · (R−1 ⊗ id) =

exp

(− + +

2

) .

Since the element + + belongs to the center of the algebraof horizontal chord diagrams it commutes with any other element of thisalgebra. In particular, the exponent of this elements commutes with Φ.

Property 3. Now let us prove the pentagon relation. We will follow aprocedure similar to example 12.3.9. Assume that the distances between thestrings of the tangles are as indicated in Figure 12.5.

Since the tangles are isotopic, we have Z(LHSt,mb,w ) = Z(RHSt,mb,w ). Hence

ε−t−w2πi·∆3( ) · ε−m−w

2πi· · Z(LHSt,mb,w ) · εm−w

2πi· · ε b−w

2πi·∆1( )

= ε−t−w2πi·∆3( ) · ε−m−w

2πi· · Z(RHSt,mb,w )ε

m−w2πi· · ε b−w

2πi·∆1( ) .

Now, following 12.3.9, we are going to insert the commuting and mutuallycanceling factors into the left and right hand sides and then pass to thelimit as ε→ 0. A possible choice of insertions for the left hand side is shownin Figure 12.6 and for the right hand side, in Figure 12.7. Values of thelimits at the right columns are justified in problems 14 and 15 at the end ofChapter 8.

Property 4, the hexagon relation, can be proven in the same spirit as thepentagon relation. We leave details to the reader as an exercise (problem 8at the end of the Chapter).

12.4.4. Rational associators. Drinfeld proved that there exists an asso-ciator with rational coefficients. In [BN2] D. Bar-Natan, following [Dr2],


gives a construction of an associator by induction on the degree. He im-plemented the inductive procedure in Mathematica ([BN5]) and computed

the logarithm of the associator up to degree 7. With the notation a = ,

b = his answer looks as follows.

log Φ = 148 [ab]− 1

1440 [a[a[ab]]]− 111520 [a[b[ab]]]

+ 160480 [a[a[a[a[ab]]]]] + 1

1451520 [a[a[a[b[ab]]]]]

+ 131161216 [a[a[b[b[ab]]]]] + 17

1451520 [a[b[a[a[ab]]]]]

+ 11451520 [a[b[a[b[ab]]]]]

−(the similar terms with interchanged a and b) + . . .

12.4.5. Remark. This expression is obtained from ΦKZ expanded to degree7, by substitutions ζ(3)→ 0, ζ(5)→ 0, ζ(7)→ 0.

12.4.6. Remark. V. Kurlin [Kur] described all group-like associators mod-ulo the second commutant. The Drinfeld associator ΦKZ is one of them (seeSec. 12.2.21).

12.4.7. The axioms do not define the associator uniquely. The next theoremdescribes the totality of all associators.

Theorem. ([Dr1, LM2]). Let Φ and Φ be two associators. Then there is

a symmetric invertible element F ∈ A(2) such that

Φ = (id⊗ F−1) ·∆2(F−1) · Φ ·∆1(F ) · (F ⊗ id) .

Conversely, for any associator Φ and an invertible element F ∈ A(2) theleft hand side of this equation is an associator.

The operation Φ 7→ Φ is called twisting by F . Diagrammatically, it looksas follows:

Φ =

F−1

∆2(F−1)

Φ

∆1(F )

F

12.4.8. Exercise. Prove that the twist by an element F = exp(α ) isidentical on any associator.


12.4.9. Exercise. Prove that the elements f ⊗ id and ∆1(F ) commute.

12.4.10. Example. Take Φ = ΦBN (the associator of the previous remark)

and Φ = ΦKZ. It is remarkable that both of these associators are horizontal,i.e. belong to the subalgebra Ah(3) of horizontal diagrams, but one can beconverted into another only by a non-horizontal twist, for example, by thefollowing series:

F =

which ensures the coincidence of results up to degree 4.

— the coresponding beautiful calculation will be added byS.D.!! —

Twisting and the previous theorem were discovered by V. Drinfeld [Dr1]in the context of quasi-triangular quasi-Hopf algebras. Then it was adaptedfor algebras of tangled chord diagrams in [LM2]

Proof. ......Sketch of the proof to be written by S.D...........

12.4.11. We can use an arbitrary associator Φ to define a combinatorialKontsevich integral ZΦ(K) corresponding to Φ replacing ΦKZ by Φ in theconstructions of section 12.3.5.

Theorem ([LM2]). For any two associators Φ and Φ the correspondingcombinatorial integrals are equal: ZΦ(K) = ZeΦ(K).

The proof is based on the fact that for any parenthesized tangle T theintegrals ZΦ(T ) and ZeΦ(T ) are conjugate in the sense that ZeΦ(T ) = Fb ·ZΦ(T ) · F−1

t , where the elements Fb and Ft depend only on the parenthesisstructures on the bottom and top of the tangle T respectively. Since for aknot K, considered as a tangle, the bottom and top are empty, the integralsare equal.

12.4.12. Corollary ([LM2]). For any knot K the coefficients of the Kont-sevich integral Z(K), expanded over an arbitrary basis consisting of chorddiagrams, are rational.

Indeed, V. Drinfeld [Dr2] (see also [BN6]) showed that there exists anassociator ΦQ with rational coefficients. According to Theorems 12.3.8 and12.4.11 Z(K) = Zcomb(K) = ZΦQ

(K). The last combinatorial integral hasrational coefficients.

Exercises 289

Exercises

(1) Find the monodromy of the reduced KZ equation (p. 258) around thepoints 0, 1 and ∞.

(2) Using the action of the permutation group Sn on the configuration spaceX = Cn \ H determine the algebra of values and the KZ equation onthe quotient space X/Sn in such a way that the monodromy gives theKontsevich integral of braids (not necessarily pure). Compute the resultfor n = 2 and compare it with Exercise 8.6.6.

(3) Prove that the associator ΦKZ is group-like.

(4) Find Zcomb(31) up to degree 4 using its presentation as tangled chorddiagrams on p. 280.

(5) Draw a diagrammatic expression for the combinatorial Kontsevich inte-gral of the knot 41 corresponding to the parenthesized presentation from12.3.4. Prove the theorem 12.3.8 for the parenthesized presentation ofthe knot 41 like in example 12.3.9.

(6) Compute the Kontsevich integral of the knot 41 up to degree 4.

(7) Prove that ε2(Φ) = 1 and the pentagon relation imply the other twoequalities for strong invertibility: ε1(Φ) = 1 and ε3(Φ) = 1.

(8) Prove that ΦKZ satisfies the hexagon relation.

(9) Prove the second hexagon relation

Φ−1 · (∆1R) · Φ−1 = (R⊗ id) · Φ−1 · (id⊗R) .

for an arbitrary associator Φ.

(10) Any associator Φ in the algebra of horizontal diagrams Ah(3) can be

written as a power series in non commuting variables a = , b = ,

c = : Φ = Φ(a, b, c).a). Prove that the lemma 12.4.3 means that Φ−1(a, b, c) = Φ(b, a, c). In

particular, for an associator Φ(A,B) with values in C〈〈A,B〉〉 (likeΦBN, or ΦKZ), Φ−1(A,B) = Φ(B,A).

b). Prove that the hexagon relation from page 284 can be written inthe form Φ(a, b, c) exp

(b+c2

)Φ(c, a, b) = exp

(b2

)Φ(c, b, a) exp

(c2

).

c). (V. Kurlin [Kur]) Prove that for horizontal associator the hexagonrelation is equivalent to the relation

Φ(a, b, c)e−a2 Φ(c, a, b)e

−c2 Φ(b, c, a)e

−b2 = e

−a−b−c2 .

d). Show that for a horizontal associator Φ,

Φ∆2(R) · Φ∆2(R) · Φ∆2(R) = exp(a+ b+ c) .


(11) Express Z(H) through ΦKZ. Is it true that the resulting power seriescontains only even degree terms?

Exercises 291

Z( )

S2

(ε 2πi

)

S2

(ε− 2πi

)

Z( )

ε 2πi

ε− 2πi

Z( )

ε 2πi

ε− 2πi

Z( )

ε 2πi

ε− 2πi

Z( )

ε 2πi

ε− 2πi

Z( )

S2

(ε 2πi

)

S2

(ε− 2πi

)

Z( )

= max

−−−−−→ε→0

id⊗max⊗id

−−−−−→ε→0

S3(Φ)⊗ id

−−−−−→ε→0

R−1 ⊗ id⊗ id

−−−−−→ε→0

R−1 ⊗ id⊗ id

−−−−−→ε→0

R−1 ⊗ id⊗ id

−−−−−→ε→0

S3(Φ−1)⊗ id

−−−−−→ε→0

id⊗min⊗id

= min

Figure 12.4. Computation of the Kontsevich integral for the trefoil


LHSt,mb,w =

εbεw

εt

εm

εm

εm

RHSt,mb,w =

εbεw

εt

εm

εm

Figure 12.5. The two sides of the pentagon relation

εb−w2πi·( + )

εm−w2πi·

Z(Tmm,b,w

)ε−

m−w2πi·

ε−b−w2πi·( + )

−−−−−→ε→0 Φ⊗ id

εb−w2πi·( + )

εm−w2πi·

Z(T2m,tb,w

)ε−

m−w2πi·

ε−t−w2πi·( + )

−−−−−→ε→0 ∆2(Φ)

εt−w2πi·( + )

εm−w2πi·

Z(T t,mm,w

)ε−

m−w2πi·

ε−t−w2πi·( + )

−−−−−→ε→0 id⊗ Φ

Figure 12.6. Inserted factors for the LHS of the pentagon relation

Exercises 293

εb−w2πi·( + )

εm−w2πi·

Z(T1m,mb,w

)ε−

m−w2πi·

ε−m−w2πi·

−−−−−→ε→0 ∆1(Φ)

εm−w2πi·

εm−w2πi·

Z(T3t,mm,w

)ε−

m−w2πi·

ε−t−w2πi·( + )

−−−−−→ε→0 ∆3(Φ)

Figure 12.7. Inserted factors for the RHS of the pentagon relation

Chapter 13

Gauss diagrams

In this chapter we will discuss some combinatorial objects related to Vas-siliev invariants and differing from the objects studied before in that theycontain oriented edges (chords). We start with the notion of Gauss diagramwhich contains the complete information about the plane diagram of a knot.Gauss diagrams coming from plane knot diagrams form a proper subset ofall Gauss diagrams. So we can extend the notion of knot to virtual knotsunderstanding them simply as Gauss diagrams modulo the correspondingReidemeister moves. Various knot invariants can be described in terms ofthe Gauss diagram of a knot (and therefore generalized to virtual knots). Wegive such a description for the Jones polynomial. Then we concentrate onthe Polyak-Viro construction of Vassiliev invariants using Gauss diagrams.

13.1. Gauss diagrams and virtual knots

13.1.1. Definition. A Gauss diagram is a chord diagram with orientedchords and with numbers +1 or −1 assigned to each chord.

The three structures that constitute the notion of a Gauss diagram (thearrangement of chords, their orientations and the signs) are all independentfrom one another.

To a knot projection (a plane knot diagram) there corresponds a Gaussdiagram whose outer circle is the parameterizing circle S1, a chord is drawnfor each double point of the diagram, each chord is oriented from the overpassto the underpass and the local writhe number is assigned to each doublepoint (chord).

Here is an example of an oriented plane knot diagram and the corre-sponding Gauss diagram:

295

296 13. Gauss diagrams

1

2

4

12

3

4

21

4

3

UO

U

O

UU

O

O

+

−

−

+

3

In this picture and in the following ones we have marked a base point on thecircle which is required in the constructions of Vassiliev invariants below.

The three structures of a Gauss diagram arising from a knot projectionare interrelated. First of all, the underlying chord diagram cannot be arbi-trary, then the distribution of signs and the distribution of arrows are bothrelated between themselves and affected by the arrangement of chords.

13.1.2. Exercise. Prove that in the Gauss diagram of any knot projection

• every chord meets an even number of other chords,

• if the intersection graph of the diagram is connected, then the ar-rows determine the signs uniqely up to a common change of allsigns, and vice versa,

• if the intersection graph is disconnected, then the previous propertyholds in each connected component.

The Reidemeister moves for knot diagrams (section 1.3) induce the fol-lowing moves on Gauss diagrams (see the details in [Oll]).

V Ω1 :ε ε

V Ω2 : ε −ε −ε ε

V Ω3 :+

+−

+ −+

− +−− +

−

It is known since Gauss that the planarity of a plane curve implies somerestrictions on the arrangement of its double points (see, for example, [CE]).So not every Gauss diagram can be obtain from a planar knot diagram. Thenwe can generalize the notion of a knot.

Definition. A virtual knot is a Gauss diagram considered up to theReidemeister moves V Ω1, V Ω2, V Ω3.

13.2. The Jones polynomial via Gauss diagrams 297

Virtual knots were introduced by L. Kauffman [Ka5], and independentlyby M. Goussarov, M. Polyak, O. Viro [GPV]. They proved that many knotinvariants can be extended to invariants of virtual knots.

13.2. The Jones polynomial via Gauss diagrams

The description of the Jones polynomial given in this section is essentialy areformulation of the construction from a remarkable paper [Zul] by L. Zulli.

Let G be a Gauss diagram of a plane diagram of a knot K. Denoteby [G] the set of chords of G. The sign sign(c) of a chird c ∈ [G] can beconsidered as a value of the function sign : [G]→ −1,+1. A state s for Gis an arbitrary function s : [G] → −1,+1. So for a Gauss diagram withn chords there are 2n states. The function sign(·) is one of them. Witheach state s we associate an immersed plane curve in the following way. Wedouble every chord c according to the rule:

c

, if s(c) = 1

, if s(c) = −1

Let |s| denote the number of connected components of the curve obtainedby doubling all the chords of G (compare with the definition of soN -weightsystem in Sec.7.1.7). Also for a state s we define an integer

p(s) :=∑

c∈[G]

s(c) · sign(c) and the writhe w(s) :=∑

c∈[G]

sign(c).

The defining relations for the Kauffman bracket from Sec. 2.4 lead tothe following expression for the Jones polynomial.

Theorem.

J(K) = (−1)w(K)t3w(K)/4∑

s

t−p(s)/4(−t−1/2 − t1/2

)|s|−1,

where the sum is taken over all 2n states for G.

The same formula works for the Jones polynomial of a virtual knot aswell.


Example. For the left trefoil knot 31 we have the following Gaussdiagram.

1

3 2 G =

1

1

2

2

3 3

−− − w(G) = −3

There are eight states for such a diagram. Here are the corresponding curvesand numbers |s|, p(s).

|s|=2

p(s)=−3

|s|=1

p(s)=−1

|s|=1

p(s)=−1

|s|=1

p(s)=−1

|s|=2

p(s)=1

|s|=2

p(s)=1

|s|=2

p(s)=1

|s|=3

p(s)=3

Therefore,

J(31) = −t−9/4(t3/4(−t−1/2 − t1/2

)+ 3t1/4 + 3t−1/4

(−t−1/2 − t1/2

)

+t−3/4(−t−1/2 − t1/2

)2)

= −t−9/4(−t1/4 − t5/4 − 3t−3/4 + t−3/4

(t−1 + 2 + t

))

= t−1 + t−3 − t−4 ,

as wehad before in Chapter 2.

13.3. Arrow diagram formulas for Vassiliev invariants

M. Polyak and O. Viro suggested [PV1] the following approach to representVassiliev invariants in terms of Gauss diagrams (and thus extend Vassilievinvariants to virtual knots).

13.3.1. Definition. An arrow diagram is a based chord diagram with ori-ented chords supplied with multiplicities 1 or 2. (In the pictures, chords ofmultiplicity 2 are shown by dotted lines and the base point, by a cross.)

Here is an example of an arrow diagram:

13.3. Arrow diagram formulas for Vassiliev invariants 299

13.3.2. Definition. Let A be an arrow diagram and G a Gauss diagram,both with base points. Denote by [A], [G] the corresponding sets of chordsand by E(A), E(G) the sets of their endpoints with base point added. Sinceevery chord knows its beginning and its end, any mapping φ : [A] → [G]

induces a mapping φ : E(A) → E(G). An injective mapping φ : [A] → [G]is called a homomorphism from A to G (written φ ∈ Hom(A,G)), if the

corresponding φ preserves the cyclic order of the points.

Examples. There is exactly one homomorphism from to +−

+−

,

shown by thick chords in this picture:

+

−

−

+

There are no homomorphisms from to +−

+−

.

13.3.3. Definition. The pairing between a based arrow diagram and abased chord diagram is defined by

〈A,G〉 =∑

φ∈Hom(A,G)

∏

c∈[A]

sign(φ(c))µ(c),

where µ denotes the multiplicity (1 or 2) of the chord.

We want to use this pairing to define knot invariants by choosing anarrow diagram and then setting K 7→ G(K) 7→ 〈A,G(K)〉. In order thatthis invariant be well defined, then result must not depend on the choice ofthe Gauss diagram G(K) and the base point on it. For example, this is the

case for the arrow diagram A = (see next section).

In general, if you take an arbitrary Gauss diagramA, the value 〈A,G(K)〉

is not uniquely defined by the knot K. For example, 〈 , G(K)〉 can

change when a different base point or a different plane projection of K inused. It is true, nevertheless, that there are many linear combinations of


arrow diagrams that yield knot invariants by this construction. Moreover,a general theorem due to M. Goussarov [G3, GPV] states that any Vas-siliev invariant can be obtained from a suitable linear combination of arrowdiagrams.

13.3.4. Proposition. Suppose that A is a linear combination of arrow di-agrams with no more than n arrows such that the pairing 〈A,G(K)〉 doesnot depend on the choice of the Gauss diagram and its basepoint. Then itdefines a Vassiliev invariant of degree no greater than n.

Proof. Let A =∑

i λiAi, where Ai is an arrow diagram with ni arrows, letK be a singular knot with n+ 1 double points. Write the resolution of thesingular knot into nonsingular knots like in 3.1.1:

K =∑

ε

(−1)|ε|Kε,

where ε is a sequence of ±1’s of length (n+ 1). Each Kε| has the same setof chords, differing only by directions and signs.

The expression 〈A,G(K)〉 is a triple sum∑

i

λi∑

s

∑

ε

(−1)|ε|〈Ai, G(Kε)〉,

where the middle summation is over all subsets s of ni chords in the Gaussdiagrams G(Kε). Now we fix the number i in the first summation and thesubset s of chords in the second summation and try to find the sum overε. Since the number of chords in s is strictly smaller than (n + 1), there isat least one double point in the diagram of K which does not belong to s.Each summand with a positive resolution of this double point cancels withthe corresponding summand for the negative resolution, and the result is 0.

This completes the proof.

13.3.5. Formulas for invariants of small degree.

Degree 1. A nontrivial knot invariant of degree 1 exists only for framedknots: it is the self-linking number which may be expressed by the Gaussdiagram formula

.......................

Degree 2. The formula

v2(G) = 〈 , G〉,

13.3. Arrow diagram formulas for Vassiliev invariants 301

for the Vassiliev invariant of degree 2 taking value 0 on the unknot and value1 on the left trefoil (equal to the coefficient c2 of Conway’s polynomial), wasproved in Chapter 3, page 62.

Degree 3. Up to multiplication by a constant and modulo the invariantsof smaller degree there is only one invariant of degree 3 for unframed knots,namely the coefficient j3(K) in the power series expansion of the Jonespolynomial (see Sec. 3.7,14.2.11.2). The following two formulas for j3(K)were found by M. Polyak, O. Viro [PV1] and by S. Tyurina [Tu1].

j3(K) = 〈−3 − 6 , G(K)〉

= −6〈 *+

*+ * +

*+ * , G(K)〉

Other combinatorial formulas for v2(K) and j3(K) were found by J. Lannes[Lan].

Degree 4. There are three invariants of degree 4. A Gauss diagramformula for one of them was found in [PV1]:

v4,1(K) = 〈2 + 3 + + 2 + + 6 + 2 + 2

+ + + 2 − + + + + , G(K)〉 .This is even (invariant under the mirror reflection) additive invariant whichtakes value 3 on the trefoil 31, and value 2 on figure eight knot 41. Notethat this is the first case when we need the chords of multiplicity 2 in arrowdiagrams.

S. Tyurina found two other invariants in [Tu2]. For the appropriatechoices of weight systems Tyurina’s formulas may be rewritten as follows

v4,2(K) = 〈−2 − 2 − 6 − 6 + 2 + 2 + 2

+2 + + + + + + , G(K)〉 .

v4,3(K) = 〈− − − 2 − 2 + + +

+ + + + + + , G(K)〉 .


13.4. The algebra of arrow diagrams

The vector space spanned by all arrow diagrams can be made into an algebrain the following way. We will define it in a more general context of stringlinks (the case of arrow diagrams for knots considered before is a particularcase).

13.4.1. Definition. A (string link) arrow diagram is a collection of n ori-ented intervals with pairs of distinct internal points connected by orientedchords.

With a plane diagram of a string link one can naturally associate anarrow diagram. Here is an example:

Arrow diagrams can be multiplied by putting one on top of the other,just like the tangled chord diagrams (see Sec. 8.6.2). For arrow diagrams, thecounterpart of the 4-term relation is the 6-term relation defined as follows:

− + − + − = 0.

13.4.2. Definition. The algebra of arrow diagrams on n strings ~An is thequotient of the space generated by all arrow diagrams modulo the 6T rela-tions.

Setting = + , one gets a mapping from the algebra of

chord diagrams on n strings An into the algebra ~An, because 4T relationsgo into some linear combinations of 6T relations.

Conjecture. (1) This mapping is injective. (2) Its image coincides with

the symmetric part of ~An.

The importance of the algebra of arrow diagrams is based on the follow-ing three facts:

(1) The algebra ~An provides a convenient means to work with Gaussdiagram formulas for Vassiliev invariants.

Exercises 303

(2) It is the counterpart of the algebra of chord diagrams for virtuallinks.

(3) The 6 term relation is closely connected with the classical Yang-Baxter equation.

For the details we refer the reader to the paper of M. Polyak [Po].

Exercises

(1) Three invariants of order 4 come from coefficients of the Conway andJones polynomials (see Sec.3.7). Namely c2 · c2, c4, and j4. Expressthem in terms of invariants v4,1, v4,2, and v4,3 from Sec. 13.3.5.

(2) Let A be the algebra of chord diagrams, while ~A be the algebra oforiented (arrow) chord diagrams. Prove that the natural mapping A →~A is well-defined, i. e. that the 6T relation implies the 4T relation.

(3) Let ~A be the algebra of arrow diagrams, while ~C be the algebra ofacyclic arrow graphs (oriented closed diagrams). Prove that the natural

mapping ~A → ~C is well-defined, i.e. that the ~STU relation implies the6T relation.

(4) ∗ Define an analogue of the algebra of oriented closed diagrams ~C gen-

erated by all graphs, not only acyclic. Denote it by ~Call. Is it true that

the natural mapping ~C → ~Call is an isomorphism?

(5) ∗ Construct an analogue of the algebra of open diagrams B consistingof graphs with oriented edges.

Chapter 14

Miscellany

14.1. The Melvin–Morton conjecture

14.1.1. Formulation. Roughly speaking the Melvin–Morton conjecturesays that the Alexander-Conway polynomial can be read from the highestorder part of the colored Jones polynomial.

According to exercise (27) of Chapter 7 (see also [MeMo, BNG]) thecoefficients Jkn of the unframed colored Jones polynomial Jk (Section 10.4.3)are polynomials in k of degree at most n+1 without free terms. So we maywrite

Jknk

=∑

0≤j≤nbn,jk

j andJk

k=∞∑

n=0

∑

0≤j≤nbn,jk

jhn ,

where bn,j are Vassiliev invariants of order 6 n. Under the highest orderpart of the colored Jones polynomial we understood the Vassiliev powerseries invariant

MM :=

∞∑

n=0

bn,nhn .

The Melvin–Morton conjecture. [MeMo] The highest order part ofthe colored Jones polynomial MM is inverse to the version of the Alexander–Conway power series A defined by equations (1-2). In other words,

MM(K) ·A(K) = 1

for any knot K.

305

306 14. Miscellany

14.1.2. Historical remarks. In [Mo] H. Morton proved the conjecturefor torus knots. After this L. Rozansky [Roz1] proved the Melvin–Mortonconjectures on a level of rigor of Witten’s path integral interpretation ofJones invariant. The first complete proof was done by D. Bar-Natan andS. Garoufalidis [BNG]. They invented a remarkable reduction of the con-jecture to a certain equation for weight systems via canonical invariants. Wewill review this reduction in Section 14.1.3. They checked the equation usingcalculations of the weight systems on chord diagrams. Also they proved amore general theorem [BNG] relating the highest order part of an arbitraryquantum invariant to the Alexander-Conway polynomial. Following [Ch2]we shall represent another proof of this generalized Melvin–Morton conjec-ture in Section 14.1.7. A. Kricker, B. Spence and I. Aitchison [KSA] provedthe Melvin–Morton conjecture used the cabling operations described in Sec-tion 9.5. Later, developing this method, A. Kricker [Kri1] also proved thegeneralization. In the paper [Vai1] A. Vaintrob gives another proof of theMelvin–Morton conjectures. He used calculations on chord diagrams andthe Lie superalgebra gl(1|1) which is responsible for the Alexander–Conwaypolynomial. An idea to use the restriction of the equation for weight systemsto the primitive space was explored in [Ch1, Vai2]. We shall follow [Ch1]in the direct calculation of the Alexander–Conway weight system in Section14.1.5.

B. I. Kurpita and K. Murasugi found a different proof of the Melvin–Morton conjecture which does not use Vassiliev invariants and weight sys-tems [KuM].

The works on the Melvin–Morton conjecture stimulated L. Rozansky[Roz2] to formulate an exiting conjecture about the fine structure of theKontsevich integral. The conjecture was proved by A. Kricker [Kri2]. Weshall discuss this in Section 10.7.

14.1.3. Reduction to weight systems. Both power series Vassiliev in-variants MM and A are canonical, so does their product (exercise (7)). Theinvariant of the right-hand side that is identically equal to 1 on all knotsis also a canonical invariant. Hence the Melvin–Morton conjecture is anequality for canonical invariants. It is enough to prove the equality of theirsymbols.

Introduce the notation

SMM := symb(MM) =∞∑

n=0

symb(bn,n) ;

SA := symb(A) = symb(C) =

∞∑

n=0

symb(cn) .

14.1. The Melvin–Morton conjecture 307

The Melvin–Morton conjecture in equivalent to the relation

SMM · SA = 10 .

It is obvious in degrees 0 and 1. So basicly we have to prove that in degree> 2 the product SMM · SA equals zero. To prove this we have to prove thatSMM · SA(p1 · · · · · pn) = 0 on any product p1 · · · · · pn of primitive elementsof degree > 1.

The weight system SMM is the highest part of the weight system ϕ′Vksl2/k

from exercise (27) of the Chapter 7. The last one is multiplicative as it wasexplained in Section 7.1.4. So SMM is multiplicative too. Exercise (15) ofthe Chapter 3 implies that the weight system SA is also multiplicative. Inother words, both weight systems SMM and SA are group-like elements ofthe Hopf algebra of weight systemsW. A product of two group-like elementsis a group-like. So the weight system SMM · SA is multiplicative. Therefore,it is sufficient to prove that

SMM · SA∣∣P>1

= 0 .

By the definitions of the multiplication of weight systems and primitiveelements,

SMM · SA(p) = (SMM ⊗ SA)(δ(p)) = SMM(p) + SA(p) .

Therefore we have reduced the Melvin–Morton conjecture to the equality

SMM

∣∣P>1

+ SA∣∣P>1

= 0 .

Now we shall exploit the filtration

0 = P1n ⊆ P2

n ⊆ P3n ⊆ · · · ⊆ Pnn = Pn .

from Section 5.5.2 and the wheel wn that generates Pnn/Pn−1n for even n and

belongs to Pn−1n for odd n.

The Melvin–Morton conjecture follows from the next theorem.

14.1.4. Theorem. The weight systems SMM and SA have the properties

1) SMM

∣∣Pn−1

n= SA

∣∣Pn−1

n= 0;

2) SMM(w2m) = 2, SA(w2m) = −2.

The equation SA(w2m) = −2 is a particular case of exercise (33) ofChapter 7. Exercise (25) of the same chapter implies that for any p ∈ Pn−1

n

the degree in c of the polynomial ρsl2(p) is less than or equal to [(n− 1)/2].

After the substitution c = k2−12 corresponding to the weight systems of the

colored Jones polynomial, the degree of the polynomial ρVksl2

(p)/k in k will

be at most n − 1. Therefore, its n-th term vanishes and SMM

∣∣Pn−1

n= 0.

Also the exercise (24) there gives that the highest term of the polynomial

308 14. Miscellany

ρsl2(w2m) is 2m+1cm. Again the substitution c = k2−12 (taking the trace

of the corresponding operator and dividing the result by k) gives that the

highest term of ρVksl2

(w2m)/k will be 2m+1k2m

2m = 2k2m. Hence SMM(w2m) = 2.

The only equation which remains to prove is that SA∣∣Pn−1

n= 0. We shall

prove in teh next subsection.

14.1.5. Alexander–Conway weight system. Using the state sum for-mula for SA from the exercise (33) of the Chapter 7 we are going to provethat SA(p) = 0 for any closed diagram p ∈ Pn−1

n .

First of all note that any such p ∈ Pn−1n has an internal vertex connected

with three other internal vertices. Indeed, each external vertex is connectedwith only one internal vertex. The number of external vertices is not greaterthan n − 1. The total number of vertices of p is 2n. So there should be atleast n+ 1 internal vertices, and only n− 1 of them may be connected withexternal vertices.

Fix such a vertex connected with three other internal vertices. Thereare two possible cases: either all the three vertices are different or two ofthem coincide.

The second case is easier, so let us start with it. Here we have a “bubble”

. After resolving the vertices of this fragment and erasing the curves

with more than one component we are left with the linear combination ofcurves −2 + 2

which cancel each other. So SA(p) = 0.

For the first case we formulate our claim as a lemma.

14.1.6. Lemma. SA

( )= 0.

We shall utilize the state surfaces Σs(p) from problem 29 of Chapter 7.Neighbourhoods of “+”- and “−”-vertices of our state look on the surfacelike three meeting bands:

(1)+

;−

= = = .

So a switching of a mark (value of the state) at a vertex means thereglueing of the three bands along two chords on the surface:

cut along chords interchange glue .


Proof. We are going to divide the set of all those states s for which the statesurface Σs(p) has one boundary component in pairs such that the states sand s′ of the same pair differ by an odd number of marks. So they willcancel each other and will contribute zero to SA(p).

In fact, to do this we shall adjust only marks of the four vertices of thefragment pictured in the lemma. So the marks ε1, . . . , εl and ε′1, . . . , ε′lin the states s and s′ will be the same except for some marks of the fourvertices of the fragment. Denote the vertices by v, va, vb, vc and their marksin the state s by ε, εa, εb, εc.

Assume that Σs(p) has one boundary component. Modifying the surfaceas in (1) we can suppose that the neighbourhood of the fragment has theform

vvb

vc

vaa

b

c

b1

c2

a1

a2

b2

c1

Draw nine chords a, a1, a2, b, b1, b2, c, c1, c2 on our surface as shown onthe picture. The chords a, b, c are located near the vertex v; a, a1, a2 nearthe vertex va; b, b1, b2 near vb and c, c1, c2 near vc.

Since our surface has only one boundary component, we can draw it asa plane circle and a, a1, a2, b, b1, b2, c, c1, c2 as chords inside it. Look atthe possible chord diagrams thus obtained.

If two, say b and c, of three chords located near a vertex, say v, donot intersect, then the surface Σ...,−ε,εa,εb,εc,...(P ) obtained by switching themark ε to −ε has only one boundary component too. Indeed, the regluingeffect along two non-intersecting chords can be seen on chord diagrams asfollows:

cut along chords interchange glue .

So, in this case, the state s = . . . ,−ε, εa, εb, εc, . . . should be paired withs′ = . . . , ε, εa, εb, εc, . . . .

Therefore, the switching of a mark at a vertex increases the number ofboundary components (and then such a marked diagram may give a non-zero contribution to SA(D)) if and only if the three chords located near thevertex intersect pairwise.

310 14. Miscellany

Now we can suppose that any two of the three chords in each triple(a, b, c), (a, a1, a2), (b, b1, b2), (c, c1, c2) intersect. This leaves us with onlyone possible chord diagram:

a1abb1

a2

a1

c2

c

a

a2 c1 c2b2

b

c

c1

b1

b2

So the boundary curve of the surface connects the ends of our fragment asin the left picture below.

Σ...,ε,εa,εb,εc,...(p)

a

a

bb

cc

b1

b1

c2c2

a1

a1

a2

a2

b2

b2

c1

c1

Σ...,ε,−εa,−εb,−εc,...(p)

Switching marks at va, vb, vc gives a surface also with one boundary compo-nent as in the right picture above. Pairing the state s = . . . , ε, εa, εb, εc, . . . up with s′ = . . . , ε,−εa,−εb,−εc, . . . we get the desired result.

The Lemma and thus the Melvin–Morton conjecture are proved.

14.1.7. Generalization of the Melvin–Morton conjecture to otherquantum invariants. Let g be a semi-simple Lie algebra, Vλ be an irre-ducible representation of g of the highest weight λ. Denote by h a Cartansubalgebra of g, by R the set of all roots and by R+ the set of positive roots.Let 〈·, ·〉 be the scalar product on h∗ induced by the Killing form. These

data define the unframed quantum invariant θVλg which after the substitu-

tion q = eh and the expansion into a power series in h cab be written as (seeSection10.4.2)

θVλg =

∞∑

n=0

θλg,nhn.

Theorem. ([BNG]).1) The invariant θλg,n/dim(Vλ) is a polynomial in λ of degree at most n.


2) Define the Bar-Natan–Garoufalidis function BNG as a power series in hwhose coefficient at hn is the degree n part of the polynomial θλg,n/dim(Vλ).Then for any knot K,

BNG(K) ·∏

α∈R+

Aα(K) = 1 ,

where Aα is the following normalization of the Alexander–Conway polyno-mial:

Aα( ) − Aα( ) = (e〈λ,α〉h

2 − e−〈λ,α〉h

2 )Aα( ) ;

Aα( ) =〈λ, α〉h

e〈λ,α〉h

2 − e− 〈λ,α〉h2

.

Proof. The symbol SBNG is the highest part (as a function of λ) of the Lie

algebra weight system ϕ′Vλg associated with the representation Vλ. According

to the exercise (5) the symbol of Aα in degree n equals 〈λ, α〉nsymb(cn).

In the same way the relation between know invariants can be reducedto the relation between their symbols

SBNG∣∣Pn

+∑

α∈R+

〈λ, α〉nsymb(cn)∣∣Pn

= 0 ,

for n > 1.

As before SBNG∣∣Pn−1

n= symb(cn)

∣∣Pn−1

n= 0, and symb(cn)(w2m) = −2.

So it remains to prove that

SBNG(w2m) = 2∑

α∈R+

〈λ, α〉2m .

To prove this equality we shall use the the method of the Section 7.2.For this we first take the Weyl basis of the linear space g and write the Liebracket tensor J in this basis.

Fix the root space decomposition g = h⊕(⊕α∈R

gα). The Cartan subal-

gebra h is orthogonal to all the gα’s and gα is orthogonal to gβ for β 6= −α.Choose the elements eα ∈ gα and hα = [eα, e−α] ∈ h for each α ∈ R in sucha way that 〈eα, e−α〉 = 2/〈α, α〉, and for any λ ∈ h∗, λ(hα) = 2〈λ, α〉/〈α, α〉.

The elements hβ , eα, where β belongs to a basis B(R) of R and α ∈ R,form the Weyl basis of g. The Lie bracket [·, ·] as an element of g∗ ⊗ g∗ ⊗ g

312 14. Miscellany

can be written as follows:

[·, ·] =∑

β∈B(R)

α∈R

(h∗β ⊗ e∗α ⊗ α(hβ)eα − e∗α ⊗ h∗β ⊗ α(hβ)eα

)

+∑

α∈Re∗α ⊗ e∗−α ⊗ hα +

∑

α,γ∈R

α+γ∈R

e∗α ⊗ e∗γ ⊗Nα,γeα+γ ,

where ∗’s mean the elements of the dual basis. The second sum will be themost important because the first and third ones will not give any contribu-tion to the Bar-Natan–Garoufalidis weight system SBNG.

After identification of g∗ and g via 〈·, ·〉 we get e∗α = (〈α, α〉/2)e−α. Inparticular, the second sum of the tensor J will be

∑

α∈R

(〈α, α〉/2

)2e−α ⊗ eα ⊗ hα .

According to the Secction 7.2, to calculate SBNG(w2m) we have to asso-ciate the tensor −J with each internal vertex. Then make all contractionscorresponding to edges connecting internal vertices. After that take the

product ρVλg (w2m) of all operators in Vλ corresponding to external vertices.

ρVλg (w2m) is a scalar operator of multiplication by a certain constant. This

constant is a polynomial in λ of degree at most 2m. Its part of degree 2mwill be SBNG(w2m).

We associate the tensor −J with an internal vertex according to thecyclic ordering of the three edges in such a way that the third tensor factorof −J will correspond to the edge connecting the vertex with an externalvertex. After that we will take the product of operators corresponding tothese external vertices. This means that we will take the product of op-erators corresponding to the third tensor factor of −J . Of course, we areinterested only in those operators which are linear in λ. One can show (see,for example, [BNG, Lemma 5.1]) that it is possible to choose a basis inthe space of the representation Vλ in such a way that only Cartan operatorshα and raising operators eα (α ∈ R+) will be linear in λ while the loweringoperators e−α (α ∈ R+) will not depend on λ. So we have to take intoaccount only those summands of −J which have hα or eα (α ∈ R+) as thethird tensor factor. Further, to calculate the multiplication constant of ourproduct it is sufficient to act by the operator on any vector. Let us choosethe highest weight vector v0 for this. The Cartan operators hα multiply v0by λ(hα) = 2〈λ, α〉/〈α, α〉. So indeed they are linear in λ. But the raisingoperators eα (α ∈ R+) send v0 to zero. This means that we have to takeinto account only those summands of −J whose third tensor factor is one

14.2. The Goussarov–Habiro theory 313

of the hα’s. This is exactly the second sum of J with the opposite sign:∑

α∈R

(〈α, α〉/2

)2eα ⊗ e−α ⊗ hα.

Now making all contractions corresponding to the edges connecting internalvertices of w2m we get the tensor:

∑

α∈R

(〈α, α〉/2

)2mhα ⊗ . . .⊗ hα︸︷︷︸

2m times

.

The corresponding element of U(Γ) acts on the highest weight vector v0 asmultiplication by

SBNG(w2m) =∑

α∈R〈λ, α〉2m = 2

∑

α∈R+

〈λ, α〉2m .


14.2. The Goussarov–Habiro theory

14.2.1. Formulation of the Goussarov–Habiro theorem.

14.2.1.1. In September 1995, at a conference in Oberwolfach, Michael Gous-sarov reported about a theorem describing the pairs of knots indistinguish-able by Vassiliev invariants of order ≤ n. As usual with Goussarov’s resultsthe corresponding publication [G4] appeared several years later. K. Habiroindependently found the same theorem [Ha1, Ha2]. In this chapter we willdiscuss a version of this theorem and related results. Other approaches tothe Goussarov–Habiro’s type theorems can be found in [CT, Sta3, TY].

Theorem (Goussarov–Habiro). Let K1 and K2 be two knots. Then v(K1) =v(K2) for any Z-valued Vassiliev invariant v of order ≤ n if and only if K1

and K2 are related by a finite sequence of moves Mn:

︸︷︷︸n+ 2 components


14.2.1.2. Denote by Bn (resp. Tn) the tangle on the left (resp. right) sideof the move Mn. The tangle Bn is an example of Brunnian tangles char-acterized by the property that removing any of its components makes theremaining tangle to be isotopic to the trivial tangle Tn−1 with n+1 compo-nents.

We start the series of moves Mn with n = 0:

M0 : B0 = = T0which is equivalent to the usual change of crossing.

314 14. Miscellany

14.2.2. Exercise. Represent the change of crossing move

as a composition of M0 and an appropriate Reidemeister move of secondtype.

For n = 1 the move M1 looks like

M1 : B1 = = T1

It is also known as the Borromean move

Since there are no invariants of order ≤ 1 except constants (Proposition3.2.4), the Goussarov–Habiro theorem implies that any knot can be trans-formed to the unknot by a finite sequence of Borromean movesM1, i.e. M1

is an unknotting operation.

14.2.3. Remark. Coincidence of all Vassiliev invariants of order ≤ n im-plies the coincidence of all Vassiliev invariants of order ≤ n− 1. This meansthat one can accomplish a moveMn by a sequence of movesMn−1. Indeed,let us deform the tangle Bn to the tangle on the left in the following picture:

Mn−1

(to see that this is indeed Bn, look at the picture closely and try to untanglewhat can be untangled, working from right to left). Then the tangle in thedashed rectangle is Bn−1. To perform the move Mn−1 we must cut it andpaste Tn−1 instead. This gives us the tangle on the right also containingBn−1. Now performing once more the move Mn−1 we obtain the trivialtangle Tn.

14.2.4. Reformulation of the Goussarov–Habiro theorem.

14.2.4.1. Recall that we denoted by K the set of all (isotopy classes of)knots. Consider a free Z-module Z[K] consisting of all finite formal Z-linear


combinations of knots. Then using the Vassiliev skein relation

= − ,

we can think about a singular knot simply as a short notation for the ap-propriate linear combination from Z[K]. In particular, a singular knot withn double points denotes the corresponding linear combination of 2n knotswith the coefficients ±1.

Definition. Let Kn denote the Z-submodule of Z[K] generated by singularknots with > n double points. So, a Vassiliev invariant of order ≤ n is alinear function on Z[K] which vanishes on the submodule Kn.Definition. Let Hn denote the Z-submodule of Z[K] generated by differ-ences of two knots obtained one from another by a single move Mn . Forexample, 31 − 63 belongs to H2.

14.2.4.2. Goussarov–Habiro’s theorem can be reformulated as follows.

Theorem. For all n the submodules Kn and Hn coincide.

Proof. (Of the equivalence of the two statements.) Indeed, if one knot K1

can be obtained from another one K2 by a finite sequence of moves Mn,then the difference K1 − K2 belongs to Hn. Therefore, by theorem 2.2.2,it belongs to Kn, i.e., it is equal to a linear combination of knots with > ndouble points. Hence v(K1) = v(K2) for any Vassiliev invariant v of order≤ n. Conversely, if v(K1) = v(K2) for any Vassiliev invariant v of order ≤ n,then K1 − K2 belongs to Kn and, by theorem 14.2.4.2, to Hn. ThereforeK1−K2 is equal to a sum of differences of pairs of knots obtained one fromanother by a singleMn move. This means that all summands in this sum canbe canceled except for the two knots K1 and K2. This allows us to rearrangethe pairs in this sum in such a way that K1 occurs in the first pair and −K2

occurs in the last pair and two consecutive pairs contain the isotopic knots,one with plus one and another with minus one coefficients. Obviously suchan order of pairs gives us a sequence ofMn moves that transform K1 to K2.So Goussarov–Habiro’s theorem 14.2.1.1 follows from theorem 14.2.4.2. Inthe same way theorem 14.2.4.2 follows from Goussarov–Habiro’s theorem.So they are equivalent.

To prove theorem 14.2.4.2 we have to prove two inclusions: Hn ⊆ Knand Kn ⊆ Hn. The first one is an easy task. We do it in the next section.The hard part — proof of the second inclusion — can be found in [G4,Ha1, Ha2, Sta3, TY].

14.2.5. Proof of sufficiency.

316 14. Miscellany

14.2.5.1. In this section we prove that Hn ⊆ Kn. For this it is enough torepresent the difference Bn−Tn as a linear combination of tangles with n+1double points in each. Let us choose the orientations of the components ofour tangles as shown. We are going to use the Vassiliev skein relation andgradually transform the difference Bn − Tn into the required form.

Bn − Tn = −

= + −

But the difference of the last two tangles can be expressed as a singulartangle:

= −

We got a presentation of Bn−Tn as a linear combination of two tangleswith one double point in each between the first and the second componentsof the tangles. Now we add and subtract isotopic singular tangles with onedouble point:

Bn − Tn =

−

−(

−)

Then using the Vassiliev skein relation we can see that the difference in thefirst pair of parentheses is equal to

− + −

= − +


Similarly the difference in the second pair of parentheses would be equal to

− + −

= − +

So we have represented Bn − Tn as a linear combination of four tangleswith two double points in each, one is between the first and the secondcomponents of the tangles, and another one is between the second and thirdcomponents:

Bn − Tn = − +

+ −

Continuing in the same way we come to a linear combination of 2n

tangles with n + 1 double points in each occurring between consecutivecomponents. It is easy to see that if we change the orientations of arbitraryk components of our tangles Bn and Tn, then the whole linear combinationwill be multiplied by (−1)k.

14.2.6. Example.

B2−T2 = − − +

14.2.7. Invariants of order 2.

14.2.8. Example. There is only one (up to proportionality and adding aconstant) nontrivial Vassiliev invariant of order ≤ 2. It is the coefficient c2of the Conway polynomial defined in Sec. 3.8.3.

Consider two knots

31 = , 63 = .

We choose the orientations as indicated. Their Conway polynomials

C(31) = 1 + t2 , C(63) = 1 + t2 + t4.

318 14. Miscellany

have equal coefficients of t2. Therefore for any Vassiliev invariant v of order≤ 2 we have v(31) = v(63). In this case the Goussarov–Habiro theoremstates that it is possible to obtain the knot 63 from the knot 31 by movesM2 : B2 T2

M2 :

Let us show this. We start with the standard diagram of 31, and thentransform it in order to specify the tangle B2.

31 = ∼= ∼= ∼= ∼=

∼= ∼= ∼= ∼=

∼= ∼= ∼=

∼= ∼=

Now we have the tangle B2 in the dashed oval. To perform the moveM2 we must replace it by the trivial tangle T2:

∼= ∼= ∼= ∼= = 63

14.2.8.1. The mod 2 reduction of c2 is called the Arf invariant of a knot. Adescription of the Arf invariant similar to the Goussarov–Habiro descriptionof c2 was obtained by L. Kauffman.


Theorem. (L.Kauffman [Ka1, Ka2]). K1 and K2 have the same Arf in-variant if and only if K1 can be obtained from K2 by a finite number of socalled pass-moves:

The orientations are important. Allowing pass-moves with arbitraryorientations we obtain an unknotting operation (see [Kaw2]).

14.2.8.2. Open problem. (L. Kauffman) Find a set of moves relating theknots with the same c2 modulo n, for n = 3, 4,. . . .

14.2.8.3. Open problem. Find a set of moves relating any two knots withthe same Vassiliev invariants modulo 2 (3, 4, ...) up to the order n.

14.2.8.4. Open problem. Find a set of moves relating any two knots withthe same Conway polynomial.

The paper by S. Naik and T. Stanford [NaS] might be useful in studyingthese problems. It describes the moves relating two knots with the sameSeifert matrix.

14.2.9. The Goussarov groups.

14.2.10. Definition. Two knots K1 and K2 are called n-equivalent if theycannot be distinguished by Z-valued Vassiliev invariants of order ≤ n, i.e.,v(K1) = v(K2) for any v ∈ VZ

n .

Let Gn be the set of n-equivalence classes of knots.

14.2.11. Theorem (Goussarov [G1]). The set Gn is an Abelian group withrespect to the connected sum of knots.

We call the group Gn the n-th Goussarov group.

Since there are no Vassiliev invariants of order ≤ 1 except constants, thezeroth and the first Goussarov groups are trivial. The proof of the theoremin [G1] is an explicit construction of an n-inverse to a given knot K, i.e.such a knot K−1

n that the connected sum K#K−1n has all trivial Vassiliev

invariants up to the order n. The proof proceeds by induction on n andgives the presentation of the n-inverse K−1

n as a connected sum K−1n =

K2#K3# . . .#Kn such that for any j = 2, . . . , n the truncated connectedsum K2#K3# . . .#Kj is a j-inverse to K. In spite of constructivity it isvery difficult to follow the proof even in the simplest nontrivial example ofa 2-inverse of the trefoil 31. In this section we describe the groups G2, G3.

14.2.11.1. The second Goussarov group G2. Consider our favorite invariantof order 2, the coefficient c2 of t2 in the Conway polynomial C(K) (seesection 2.3). According to exercise 3 at the end of Chapter 2, C(K1#K2) =

320 14. Miscellany

C(K1)·C(K2) and C(K) has the form C(K) = 1+c2(K)t2+. . . . These factsimply that c2(K1#K2) = c2(K1)+c2(K2). Therefore c2 is a homomorphismof G2 into Z. Since c2 is the only nontrivial invariant of order ≤ 2, thehomomorphism c2 : G2 → Z is in fact an isomorphism. So G2

∼= Z. From thetable in Sec.2.3.5 we can see that c2(31) = 1 and c2(41) = −1. This meansthat the knot 31 represents a generator of G2, and 41 is 2-inverse of 31. Theprime knots with up to 8 crossings are distributed in the second Goussarovgroup G2 as follows.

52 8788 820

62 77 47 75815

61 8689

81 48812

83 4185 118 817, ,

31 63 76813 816 188, ,

73 81982 51 72810814 821

0 1 2 3 4 5 6−1−2−3−4

, , , , ,,

, ,, , , , , , , , 710 , , , ,,

c2

14.2.11.2. The third Goussarov group G3. In order 3 we have one more Vas-siliev invariant j3, the coefficient at h3 in the power series expansion of theJones polynomial with substitution t = eh. The Jones polynomial is multi-plicative, J(K1#K2) = J(K1) · J(K2) (see exercise 4 at the end of Chapter2) and its expansion has the form J(K) = 1 + j2(K)h2 + j3(K)h3 + . . . (seeSec. 3.7). So we can write

J(K1#K2) = 1 + (j2(K1) + j2(K2))h2 + (j3(K1) + j3(K2))h

3 + . . .

In particular, j3(K1#K2) = j3(K1)+ j3(K2). According to exercise 5 at theend of Chapter 3, j3 is divisible by 6. Then j3/6 is a homomorphism fromG3 to Z. Together with c2 we get the isomorphism

G3∼= Z⊕ Z = Z2; K 7→ (c2(K), j3(K)/6)

Let us identify G3 with the integral lattice on a plane. The distribution ofprime knots on this lattice is shown in Figure 14.1 (numbers with a bar referto knots that differ from the corresponding table knots (see page 27) by amirror reflection).

In particular, the 3-inverse of the trefoil 31 can be represented by 62, or77. Also we can see that 31#41 is 3-equivalent to 82. Therefore 31#41#82

is 3-equivalent to the unknot, and 41#82 also represents the 3-inverse to 31.The knots 63 and 82 represent the standard generators of G3.

14.2.11.3. Open problem. Is there any torsion in the group Gn?

14.3. Bialgebra of intersection graphs

It turns out that the natural mapping that assigns to every chord diagramits intersection graph, can be converted into a homomorphism of bialgebrasγ : A → L, where A is the algebra of chord diagrams and L is an alge-bra generated by graphs modulo certain relations which was introduced byS. Lando [Lnd]. Here is his construction.

14.3. Bialgebra of intersection graphs 321

Let G be the graded vector space (over a field F) spanned by all simplegraphs (without loops nor multiple edges) as free generators:

G = G0 ⊕G1 ⊕G2 ⊕ . . . ,It is graded by the order (the number of vertices) of a graph. This space iseasily turned into a bialgebra:

1. The multiplication is defined as the disjoint union of graphs, thenextended by linearity. The empty graph plays the role of the unit in thisalgebra.

2. The comultiplication is defined similarly to the comultiplication inthe bialgebra of chord diagrams. If G is a graph, let V = V (G) be the setof its vertices. For any subset U ⊂ V denote by G(U) the graph with theset of vertices U and those vertices of the graph G whose both endpointsbelong to G. We put by definition:

(1) δ(G) =∑

U⊆V (G)

G(U)⊗G(V \ U),

and extend by linearity to the whole of G.

The sum in (1) is taken over all subsets U ⊂ V and contains as many as

2#(V ) summands.

Example of a coproduct.

δ( s s s) = 1 ⊗ s s s + 2 s ⊗ s s + s ⊗ s s+ s s ⊗ s + 2 s s ⊗ s + s s s ⊗ 1

Exercise. Check the axioms of a Hopf algebra for G.

The mapping from chord diagrams to intersection graphs does not ex-tend to a linear operator A → G, because the combinations of graphs thatcorrespond to 4-term relations for chord diagrams, do not vanish in G. Toobtain a linear mapping, it is necessary to mod out the space G by relations,consistent with the 4 term relations. Here is an appropriate definition.

Let G be an arbitrary graph and u, v an ordered pair of its vertices.The pair u, v defines two transformations of the graph G: G 7→ G′uv and

G 7→ Guv. Both graphs G′uv and Guv have the same set of vertices as G.They are obtained as follows.

If uv is an edge in G, then the graph G′uv is obtained from G by deletingthe edge uv; otherwise this edge should be added (thus, G 7→ G′uv is togglingthe adjacency of u and v).

322 14. Miscellany

The graph Guv is obtained from G in a more tricky way. Consider allvertices w ∈ V (G) \ u, v which are adjacent in G with v. Then in the

graph Guv vertices u and w are joined by an edge if and only if they arenot joined in G. For all other pairs of vertices their adjacency in G and in

Guv is the same. Note that the two operations applied at the same pair ofvertices, commute and hence the graph G′uv is well-defined.

14.3.1. Definition. A four-term relation for graphs is

(2) G−G′uv = Guv − G′uvExample.

u u uv v v vu

− = −

Exercises. 1. Check that, passing to intersection graphs, the four-termrelation for chord diagram carries over exactly into this four-term relationfor graphs. 2. Find the four-term relation of chord diagrams which is thepreimage of the relation shown in the previous example.

14.3.2. Definition. The graph algebra of Lando L is the quotient of thegraph algebra G over the ideal generated by all 4-term relations (2).

14.3.3. Theorem. The multiplication and the comultiplication defined aboveinduce a bialgebra structure in the quotient space L.

Proof. The only thing that needs checking is that the multiplication and thecomultiplication both respect the 4-term relation 2. For the multiplication(disjoint union of graphs) this statement is obvious. In order to verify it forthe comultiplication, it is sufficient to consider two different cases. Namely,let u, v ∈ V (G) be two distinct vertices of a graph G. The right handside summands in the comultiplication formula (1) split into two groups:those where both vertices u and v belong either to the subset U ⊂ V (G) orto its complement V (G) \ U ; and those where u and v belong to differentsubsets. By cleverly grouping the terms of the first kind for the coproduct

δ(G−G′uv−Guv+G′uv) they all cancel out in pairs. The terms of the secondkind can be paired to mutually cancel already for each of the two summands

δ(G−G′uv) and δ(Guv − G′uv). The theorem is proved.

Relations 2 are homogeneous with respect to the number of vertices,therefore L is a graded algebra. By Theorem A.2.24 (p. 346) the algebra Lis polynomial over its space of primitive elements.

Now we have a well-defined bialgebra homomorphism

γ : A → L

14.4. Estimates for the number of Vassiliev knot invariants 323

that extends the assignment of the intersection graph to a chord diagram. Itis defined by the linear mapping between the corresponding primitive spacesP (A)→ P (L).

According to S. Lando [Lnd], the dimensions of the homogeneous com-ponents of P (L) are known up to degree 7. It turns out that the ho-momorphism γ is an isomorphism in degrees up to 6, while the mappingγ : P7(A)→ P7(L) has a 1-dimensional kernel. See [Lnd] for further detailsand open problems related to the algebra L.

14.4. Estimates for the number of Vassiliev knot invariants

The knowledge of dimPi for i ≤ d is equivalent to the knowledge of dimAi,i ≤ d, or dimVi, i ≤ d. At present, the exact asymptotic behavior of thesenumbers as d tends to infinity is not known. Below, we summarize theknown results on the exact enumeration as well as on the asymptotic lowerand upper bounds.

14.4.1. Historical remarks.

14.4.1.1. Exact results. !! Add comments concerning the techniques of BN and Kneissler !!

The exact dimensions of the spaces related to Vassiliev invariants areknown up to n = 12. The results are shown in the table below. They wereobtained by V. Vassiliev for n ≤ 5 in 1990 [Va2], then by D. Bar-Natan forn ≤ 9 in 1993 [BN1] and by J. Kneissler (for n = 10, 11, 12) in 1997 [Kn0].

n 0 1 2 3 4 5 6 7 8 9 10 11 12

dimPn 0 0 1 1 2 3 5 8 12 18 27 39 55

dimAn 1 0 1 1 3 4 9 14 27 44 80 132 232

dimVn 1 1 2 3 6 10 19 33 60 104 184 316 548

14.4.1.2. Upper bounds. !! add comments concerning CD and Stoimenow–Zagier !!

A priori it was obvious that dimAd < (2d − 1)!! = 1 · 3 · · · (2d − 1),because this is the total number of chord diagrams before factorization overthe rotations.

Then, there appeared five papers where this estimate was successivelyimproved:

(1) (1993) Chmutov and Duzhin [CD1] proved that dimAd < (d− 1)!

(2) (1995) K. Ng in [Ng] replaced (d− 1)! by (d− 2)!/2.

(3) (1996) A. Stoimenow [Sto1] proved that dimAd grows slower thand!/ad, where a = 1.1.

(4) (2000) B. Bollobas and O. Riordan [BR1] obtained the asymptot-ical bound d!/(2 ln(2) + o(1))d (approximately d!/1.38d).

324 14. Miscellany

(5) (2001) D. Zagier [Zag1] improved the last result to 6d√d·d!

π2d , which

is asymptotically smaller than d!/ad for any constant a < π2/6 =1.644...

14.4.1.3. Lower bounds. In the story of lower bounds for the number ofVassiliev knot invariants there is a funny episode. The first paper by Kont-sevich about Vassiliev invariants ([Kon1], section 3) contains the followingpassage:

“Using this construction 1, one can obtain the estimate

dim(Vn) > ec√n, n→ +∞

for any positive constant c < π√

2/3 (see [BN1a], Exercise 6.14).”

Here Vn is a slip of the pen, instead of Pn, because of the referenceto Exercise 6.14 where primitive elements are considered. Exercise 6.14was present, however, only in the first edition of Bar-Natan’s preprint andeliminated in the following editions as well as in the final published versionof his text [BN1]. In [BN1a] it reads as follows (page 43):

“Exercise 6.14. (Kontsevich, [24]) Let P≥2(m) denote the number ofpartitions of an integer m into a sum of integers bigger than or equal to 2.Show that dimPm ≥ P≥2(m+ 1).

Hint 6.15. Use a correspondence like

4 3 2 2 - 10 + 1 = 4 + 3 + 2 + 2,

and . . . ”

The reference [24] was to “M. Kontsevich. Private communication.”!Thus, both authors referred to each other, and none of them gave any proof.Later, however, Kontsevich explained what he had in mind (see item 5below).

Arranged by time, the history of world records in asymptotic lowerbounds for the dimension of the primitive space Pd looks as follows.

(1) (1994) dimPd ≥ 1 (“forest elements” found by Chmutov, Duzhinand Lando [CDL3]).

(2) (1995) dimPd ≥ [d/2] (given by colored Jones function — seeMelvin–Morton [MeMo] and Chmutov–Varchenko [CV]).

(3) (1996) dimPd & d2/96 (see Duzhin [Du1]).

(4) (1997) dimPd & dlog d, i. e. the growth is faster than any polyno-mial (Chmutov–Duzhin [CD2]).

1Of Lie algebra weight systems.


(5) (1997) dimPd > eπ√d/3 (Kontsevich [Kon2]).

(6) (1997) dimPd > eC√d for any constant C < π

√2/3 (Dasbach

[Da3]).

14.4.2. Proof of the lower bound. We will sketch the proof of the lowerbound for the number of Vassiliev knot invariants, following our paper [CD2]and then explain how O. Dasbach [Da3], using the same method, managedto improve the estimate and establish the bound which is still (2003) thebest.

The idea of the proof is simple: we construct a large family of opendiagrams whose linear independence in the algebra B follows from the lin-ear independence of the values on these diagrams of a certain polynomialinvariant P , which is obtained from the universal glN invariant by means ofa simplification.

As we know from Chapter 7, the glN invariant ψglN, evaluated on an

open diagram C, is a polynomial in the generalized Casimir elements x0, x1,..., xN . This polynomial is homogeneous in the sense of the grading definedby setting deg xm = m. However, it is in general not homogeneous if xm’sare considered as simple variables of degree 1.

14.4.3. Definition. The polynomial invariant P : B → Z[x0, ..., xN ] is thehighest homogeneous part of ψglN

if all the variables are taken with degree1.

For example, if we had ψglN(C) = x2

0x2−x21, then we would have P (C) =

x20x2.

Now we introduce the family of primitive open diagrams whose linearindependence we shall prove.

14.4.4. Definition. The baguette diagram Bn1,...,nkis

Bn1,...,nk=

︸︷︷︸n1 vertices

︸︷︷︸n2 vertices

︸︷︷︸nk−1 vertices

︸︷︷︸nk vertices

. . . . . . . . . . . .

. . .

It has a total of 2(n1 + · · ·+ nk + k− 1) vertices, out of which n1 + · · ·+ nkare univalent.

To write down the formula for the value P (Bn1,...,nk), we will need the

following definitions.

326 14. Miscellany

14.4.5. Definition. A two-line scheme of order k is a combinatorial objectdefined as follows. Consider k pairs of points arranged in two rows like

. Choose one of 2k−1 subsets of the set 1, . . . , k − 1. If s belongsto the chosen subset then we connect the lower points of s-th and (s+1)-thpairs, otherwise we connect the upper pointss.

Example. Here is a scheme corresponding to k = 5 and the subset 2, 3:

.

The number of connected components in a scheme of order k is k + 1.

A weight is two sets of non-negative integers: 0 ≤ i1 ≤ n1, . . . , 0 ≤ ik ≤nk. With a weighted scheme, one can associate a monomial in the variablesxi as follows.

14.4.6. Definition. Given a weighted two-line scheme σ, we assign is tothe lower vertex of the s-th pair of σ and js = ns− is — to the upper vertex.For example

0

1

2

3

4

5

6

7

8

9

.

Then the corresponding monomial is xσ0xσ1 . . . xσkwhere σt is the sum of

integers assigned to the vertices of t-th connected component of σ.

Example. For the above scheme we get the monomial x0x4x12x5x16x8.

Now the formula for P can be stated as follows.

14.4.7. Proposition. If N > n1 + · · ·+ nk then

PglN (Bn1,...,nk) =

∑

i1,...,ik

(−1)j1+···+jk(n1

i1

). . .

(nkik

)∑

σ

xσ0xσ1 . . . xσk,

where the external sum ranges over all integers i1, . . . , ik such that 0 ≤ i1 ≤n1, . . . , 0 ≤ ik ≤ nk; the internal sum ranges over all 2k−1 schemes andxσ0xσ1 . . . xσk

is the monomial associated with the scheme σ and integersi1, . . . , ik.

Example. For the baguette diagram B2 we have k = 1, n1 = 2. Thereis only one scheme: qq . The corresponding monomial is xi1xj1 , and

P (B2) =

2∑

i1=0

(−1)j1(

2

i1

)xi1xj1

= x0x2 − 2x1x1 + x2x0 = 2(x0x2 − x21)

which agrees with the example given in Chapter 7 on page 174.


Sketch of the proof of Proposition 14.4.7. The open diagramBn1,...,nk

has k parts separated by k−1 walls. Each wall is an edge connectingtrivalent vertices to which we will refer as wall vertices. The s-th part hasns outgoing legs. We will refer to the corresponding trivalent vertices as legvertices.

The proof consists of three steps.

At the first step we study the effect of resolutions of the wall vertices.We prove that the monomial obtained by certain resolutions of these verticeshas the maximal possible degree if and only if for each wall both resolutionsof its vertices have the same sign. These signs are related to the abovedefined schemes in the following way. If we take the positive resolutions atboth endpoints of the wall number s, then we connect the lower verticesof the s-th and the (s + 1)-th pairs in the scheme. If we take the negativeresolutions, then we connect the upper vertices.

At the second step we study the effect of resolutions of leg vertices. Weshow that the result depends only on the numbers of positive resolutions ofleg vertices in each part and does not depend on which vertices in a partwere resolved positively and which negatively. We denote by is the numberof positive resolutions in part s. This yields the binomial coefficients

(ns

is

)in

the formula of Proposition 14.4.7. The total number j1 + · · ·+ jk of negativeresolutions of leg vertices gives the sign (−1)j1+···+jk .

The first two steps allow us to consider only those cases where the resolu-tions of the left is leg vertices in the part s are positive, the rest js resolutionsare negative and both resolutions at the ends of each wall have the samesign. At the third step we prove that such resolutions of wall vertices leadto monomials associated with corresponding schemes according to definition14.4.6.

We will make some comments only about the first step, because it isexactly at this step where Dasbach found an improvement of the originalargument of [CD2].

Let us fix certain resolutions of all trivalent vertices of Bn1,...,nk. We

denote the obtained T -diagram (see p. 168) by T . It consists of n = n1 +· · · + nk pairs of points and a number of lines connecting them. After asuitable permutation of the pairs T will look like a disjoint union of certainxm’s. Hence it defines a monomial in xm’s which we denote by m(T ).

Let us close all lines in the diagram by connecting the two points inevery pair with an additional short line. We obtain a number of closedcurves, and we can draw them in such a way that they have 3 intersectionpoints in the vicinity of each negative resolution and do not have other

328 14. Miscellany

intersections. Each variable xm gives precisely one closed curve. Thus thedegree of m(T ) is equal to the number of these closed curves.

Consider an oriented surface S which has our family of curves as itsboundary (the Seifert surface):

= .

The degree of m(T ) is equal to the number of boundary components b of S.The whole surface S consists of an annulus corresponding to the big circlein Bn1,...,nk

and k− 1 bands corresponding to the walls. Here is an example:

where each of the two walls on the left has the same resolutions at itsendpoints, while the two walls on the right have different resolutions attheir endpoints. The resolutions of the leg vertices do not influence thesurface S.

The Euler characteristic χ of S can be easily computed. The surface S iscontractible to a circle with k−1 chords, thus χ = 2(k−1)−(k−1+2k−2) =−k+ 1. On the other hand χ = 2− 2g− b, where g and b are the genus andthe number of boundary components of S. Hence b = k+1−2g. Therefore,the degree of m(T ), equal to b, attains its maximal value k+1 only for suchresolutions which give a zero genus surface S.

We claim that if there exists a wall whose ends are resolved with theopposite signs then the genus of S is not zero. Indeed, in this case we candraw a closed curve in S which does not separate the surface (independentlyon the remaining resolutions):

So, the contribution to P (Bn1,...,nk) is given by only those monomials

which come from equal resolutions at the ends of each wall.

Now, with Proposition 14.4.7 in hand, we can prove the following result.

14.4.8. Theorem. Let n = n1 + · · · + nk and d = n + k − 1. Baguettediagrams Bn1,...,nk

are linearly independent in B if n1, . . . , nk are all even


and satisfy the following conditions:

n1 < n2

n1 + n2 < n3

n1 + n2 + n3 < n4

· · · · · · · · · · · · · · · · · · · · ·n1 + n2 + · · ·+ nk−2 < nk−1

n1 + n2 + · · ·+ nk−2 + nk−1 < n/3.

The proof is based on the study of the supports of polynomials P (Bn1,...,nk)

— the subsets of Zk corresponding to non-zero terms of the polynomial.

Counting the number of elements described by the theorem, one arrivesat the lower bound nlog(n) for the dimension of the primitive subspace Pn ofB.

The main difficulty in the above proof is the necessity to consider 2k

resolutions for the wall vertices of a baguette diagram, which correspond tothe zero genus Seifert surface. O. Dasbach in [Da3] avoided this difficultyby considering a different family of open diagrams for which there are onlytwo ways of resolution of the wall vertices leading to the surface of minimumgenus. These are the Pont-Neuf diagrams:

PNa1,...,ak,b =

2b

a1 ak

a2

(the numbers a1, ..., ak, 2b refer to the number of legs attached to thecorresponding edge of the inner diagram).

The reader may wish to check the above property of Pont-Neuf diagramsby way of exercise. It is remarkable that Pont Neuf diagrams not only leadto a simpler considerations, but they are more numerous, too, and thus leadto a much better asymptotic estimate for dimPn. The exact statement ofDasbach’s theorem is as follows.

14.4.9. Theorem. For fixed n and k the diagrams PNa1,...,ak,b with 0 ≤a1 ≤ ... ≤ ak ≤ b, a1 + ...+ ak + 2b = 2n are linearly independent.

Counting the number of such partitions of 2n, we obtain the estimate:

330 14. Miscellany

Corollary. dimPn is asymptotically greater than ec√n for any constant

c < π√

2/3.

Exercises

(1) Show thatM1 is equivalent to the ∆ move

in the sense that, modulo Reidemeister moves, theM1 move can be ac-complished by ∆ moves and vise versa. The fact that ∆ is an unknottingoperation was proved in [Ma, MN].

(2) Prove thatM1 is equivalent to the move

(3) Prove thatM2 is equivalent to so called clasp-pass move

(4) Prove thatMn is equivalent to the move Cn:



(5) Find the inverse element of the knot 31 in the group G4.

(6) (S. Lando). Let N be a formal variable. Prove that N corankA(G) definesan algebra homomorphism L → Z[N ], where L is the graph algebraof Lando and A(G) stands for the adjacency matrix of the graph Gconsidered over the field F2 of two elements.

(7) ∗ Let λ : A → L be the natural homomorphism from the algebra ofchord diagrams into the graph algebra of Lando.• Find kerλ (unknown in degrees greater than 7).

Exercises 331

• Find imλ (unknown in degrees greater than 7).• Describe the primitive space P (L).• L is the analogue of the algebra of chord diagrams in the case of

intersection graphs. Are there any counterparts of algebras C andB?

332 14. Miscellany

8140,

82 218,

82 218, 81331816

, ,

18863,

81331816

, ,

41 817,

62 77,

62 77,

87 820,

6148

8983

81 86 85

76

52 810

51

72

815

47 75,

812

73

819

71

6148

118

118

858681

76

88

88

87 820,,

52 810

51

72

47 75,

815

819

73

71

c2

j3/6

Figure 14.1. Values of Vassiliev invariants c2 and j3 on small prime knots

Appendix

A.1. Lie algebras and their universal envelopes

A.1.1. Metrized Lie algebras. Let g be a finite-dimensional Lie algebraover C, i. e. a C-vector space equipped with a bilinear operation (commu-tator) (x, y) 7→ [x, y] subject to the rules

[x, y] = −[y, x] ,

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

An Abelian Lie algebra is an arbitrary vector space with the commutatorwhich is identical 0: [x, y] = 0 for all x, y ∈ g.

An ideal in a Lie algebra is a vector subspace stable under taking thecommutator with an arbitrary element of the whole algebra. A Lie algebrais called simple if it is not Abelian and does not contain any proper ideal.Simple Lie algebras are classified (see, for example, [FH, Hum]). Over thefield of complex numbers C there are four families of classical algebras:

Type g dim g description

An sln+1 n2 + 2n (n+ 1)× (n+ 1) matrices with zero trace, (n ≥ 1)

Bn so2n+1 2n2 + n skew-symmetric (2n+ 1)× (2n+ 1) matrices, (n ≥ 2)

Cn sp2n 2n2 + n

2n×2nmatricesX satisfying the relationXt·M+M ·X = 0,where M is the standard 2n × 2n skew-symmetric matrix

M =

O Idn

−Idn 0

, (n ≥ 3)

Dn so2n 2n2 − n skew-symmetric 2n× 2n matrices, (n ≥ 4)

and five exceptional algebras:

Type E6 E7 E8 F4 G2

dim g 78 133 248 52 14

333

334 Appendix

Apart from the low-dimensional isomorphisms

sp2∼= so3

∼= sl2; sp4∼= so5; so4

∼= sl2 ⊕ sl2; so6∼= sl4,

all the Lie algebras in the list above are different. The Lie algebra glN ofall N × N matrices is isomorphic to the direct sum of slN and the trivialone-dimensional Lie algebra C.

A representation of the Lie algebra g in the vector space V is a Liealgebra homomorphism of g into the Lie algebra End(V ) of linear operatorsin V , i.e. a mapping ρ : g→ End(V ) such that

ρ([x, y]) = ρ(x) ρ(y)− ρ(y) ρ(x).

The standard representation of a matrix Lie algebra, e.g. glN or slN , isthe one given by the identity mapping.

The adjoint representation is the action ad of g on itself according tothe rule

x 7→ adx ∈ Hom(g, g) , adx(y) = [x, y] .

It is indeed a representation, because ad[x,y] = adx · ady − ady · adx =[adx, ady].

— Add a little about representations: list of irreduciblerepresentations for sl2, Casimir tensor and Casimir num-ber! —

The Killing form, on the Lie algebra g is defined by the equality

〈x, y〉K = Tr(adxady).

Cartan’s criterion says that this bilinear form is non-degenerate if and onlyif the algebra is semi-simple, i.e. is isomorphic to the direct sum of simpleLie algebras.

A.1.2. Exercise. Prove that the Killing form is ad-invariant in the senseof the following definition.

A bilinear form 〈·, ·〉 : g⊗ g→ C is said to be ad-invariant if it satisfiesthe identity

〈adz(x), y〉+ 〈x, adz(y)〉 = 0,

or, equivalently,

(1) 〈[x, z], y〉 = 〈x, [z, y]〉.

This definition is justified by the fact described in the following exercise.

A.1. Lie algebras and their universal envelopes 335

A.1.3. Exercise. Let G be the connected Lie group corresponding to theLie algebra g and Adg : g → g its adjoint representation (see, e.g., [AdJ]).Then the ad-invariance of a bilinear form is equivalent to its Ad-invariancedefined by the natural rule

〈Adg(x),Adg(y)〉 = 〈x, y〉for all x, y ∈ g and g ∈ G.

A Lie algebra is said to be metrized, if it is equipped with an ad-invariantsymmetric non-degenerate bilinear form 〈·, ·〉 : g ⊗ g → C. This class con-tains semi-simple algebras with (a multiple of) the Killing form, AbelianLie algebras with an arbitrary non-degenerate bilinear form and their directsums.

For the classical simple Lie algebras, when their elements can be repre-sented by matrices, it is often more convenient to use, instead of the Killingform, a different bilinear form 〈x, y〉 = Tr(xy), which is proportional to theKilling form with the coefficient 1

2N for slN , 1N−2 for soN , and 1

N+2 for spN .

A.1.4. Exercise. Prove that for the Lie algebra glN the Killing form 〈x, y〉 =(Tr(adx · ady) is degenerate with defect 1 and can be expressed as follows:

〈x, y〉K = 2NTr(xy)− 2Tr(x)Tr(y) .

A.1.5. Exercise. Prove that the form Tr(xy) on glN is non-degenerate andad-invariant.

Below, we will need the following lemma.

A.1.6. Lemma. Let cijk be the structure constants of a metrized Lie algebrain a basis ei, orthonormal with respect to the ad-invariant bilinear form〈·, ·〉:

〈ei, ej〉 = δij ,

[ei, ej ] =d∑

k=1

cijkek.

Then the constants cijk are antisymmetric with respect to the permutationsof the indices i, j and k.

Proof. Equality cijk = −cjik is the coordinate expression of the fact thatthe commutator is antisymmetric: [x, y] = −[y, x]. It remains to prove thatcijk = cjki. This follows immediately from equation (1), simply by settingx = ei, y = ek, z = ej .

Corollary. Let J ∈ g⊗3 be the structure tensor of a metrized Lie algebratransferred from g∗ ⊗ g∗ ⊗ g to g⊗3 by means of the duality defined by themetric. Then J is totally antisymmetric: J ∈ ∧3g.

336 Appendix

A.1.7. Universal enveloping algebras. Let g be a finite-dimensional Liealgebra. Denote by T = ⊕n≥0Tn the tensor algebra of the vector space g.Here Tn = g⊗· · ·⊗g (n factors) is the vector space spanned by the elementsx1⊗ · · · ⊗ xn with all xi ∈ g, subject to the relations that express the lineardependence of x1 ⊗ · · · ⊗ xn on each xi. We get the universal envelopingalgebra of g, denoted by U(g), if, to these relations, we add the relationsof the form x ⊗ y − y ⊗ x − [x, y] where [x, y] is the structure bracket in g.More formally, let I be the double-sided ideal of T (g) generated by all theelements x⊗ y − y ⊗ x− [x, y], x, y ∈ g. Then by definition

U(g) = T (g)/I.

(Speaking informally, in U(g) we are allowed to treat the elements of g asassociative variables, always remembering the commutator relations thatexisted between themin g itself.)

!! add: PBW and Harish-Chandra

A.2. Bialgebras and Hopf algebras

Here we provide background information about bialgebras and Hopf algebrasnecessary for the study of the algebras of knot invariants, chord diagrams,weight systems etc.

A.2.1. Coalgebras and bialgebras. To introduce the notions of a coal-gebra and bialgebra, let us first recall what is an algebra.

A.2.2. Definition. An algebra over a field F is an F-vector spaceA equippedwith a linear mapping µ : A ⊗ A → A, called multiplication. We will onlyconsider associative algebras with a unit. Associativity means that the dia-gram

A⊗A⊗A µ⊗id−−−−→ A⊗Aid⊗µ

yyµ

A⊗A −−−−→µ

A

is commutative. The unit is a linear mapping ι : F → A (uniquely definedby the element ι(1) ∈ A) that makes commutative the diagram

F⊗A ι⊗id−−−−→ A⊗Ax

yµ

A A

where the left vertical arrow is a natural isomorphism.

A.2. Bialgebras and Hopf algebras 337

We have on purpose formulated the notion of algebra in this formal way:this allows us to derive the definition of a coalgebra by a mere reversion ofarrows.

A.2.3. Definition. A coalgebra is a vector space A equipped with a linearmapping δ : A→ A⊗A, referred to as comultiplication, and a linear mappingε : A→ F, called the counit, such that the following two diagrams commute:

A⊗A⊗A δ⊗id←−−−− A⊗Aid⊗δ

xxδ

A⊗A δ←−−−− A

F⊗A ε⊗id←−−−− A⊗Ay

xδ

A A

Algebras (resp. coalgebras) may or may not possess an additional prop-erty of commutativity (resp. cocommutativity ), defined through the follow-ing commutative diagrams:

A⊗A µ−−−−→ A

τ

x∥∥∥

A⊗A µ−−−−→ A

A⊗A δ←−−−− A

τ

y∥∥∥

A⊗A δ←−−−− A

where τ : A⊗A→ A⊗A is the permutation of the tensor factors.

The following exercise is useful to get acquainted with the notion of acoalgebra.

A.2.4. Exercise. Let ∆(n)i : A⊗n → A⊗(n+1) be defined as taking the

coproduct of the i-th tensor factor, ∆(n)0 (x) = 1⊗ x, ∆

(n)n+1(x) = x⊗ 1. Set

δn =∑n+1

i=0 (−1)i∆(n)i . Prove that the sequence δn forms a complex, i.e.

δi+1 δi = 0.

A.2.5. Definition. A bialgebra is a vector space A with the structure ofan algebra given by µ, ι and the structure of a coalgebra given by δ, ε whichagree in the sense that the following identities hold:

(1) ε(1) = 1

(2) δ(1) = 1⊗ 1

(3) ε(ab) = ε(a)ε(b)

(4) δ(ab) = δ(a)δ(b)

(in the last equation δ(a)δ(b) denotes the component-wise product in A⊗Ainduced by the product µ in A).

Note that these conditions, taken in pairs, have the following meaning:

(1,3) ⇔ ε is a (unitary) algebra homomorphism.

(2,4) ⇔ δ is a (unitary) algebra homomorphism.

338 Appendix

(1,2) ⇔ ι is a (unitary) coalgebra homomorphism.

(3,4) ⇔ µ is a (unitary) coalgebra homomorphism.

The coherence of the two structures making the definition of a bialgebracan thus be stated in either of the two equivalent ways:

• ε and δ are algebra homomorphisms,

• µ and i are coalgebra homomorphisms.

Example 1. The group algebra F[G] of a finite group G over a fieldF consists of formal linear combinations

∑x∈G λxx where λx ∈ F with the

product defined on the basic elements by the group multiplication in G.The coproduct is defined as δ(x) = x ⊗ x for x ∈ G and then extendedby linearity. Instead of a group G, in this example one can actually take asemigroup.

Example 2. The algebra FG of F-valued functions on a finite group Gwith pointwise multiplication

(fg)(x) = f(x)g(x)

and comultiplication defined by

δ(f)(x, y) = f(xy)

where the element δ(f) ∈ FG ⊗ FG is understood as a function on G × Gdue to the natural isomorphism FG ⊗ FG ∼= FG×G.

Example 3. The polynomial algebra. Let A be the symmetric algebraof a finite-dimensional vector space X, i.e. A = S(X) = ⊕∞n=0S

k(X), whereSk(X) is the symmetric part of the k-th tensor power of X. Then A is a bial-gebra with the coproduct defined on the elements x ∈ X = S1(X) ⊂ A bysetting δ(x) = 1⊗x+x⊗1 and then extended as an algebra homomorphismto the entire A.

Example 4. Universal enveloping algebras. Let g be a Lie algebra,A = U(g) — its universal enveloping algebra with usual multiplication (seesection A.1.7 for a definition and basic facts). Define δ(g) = 1⊗g+g⊗1 forg ∈ g and extend it to all of A by the axioms of bialgebra. If g is Abelian,then this example is reduced to the previous one.

Exercise. Define the appropriate unit and counit in each of the aboveexamples.

A.2.6. Primitive and group-like elements. In a bialgebra, there aretwo remarkable classes of elements: primitive elements and group-like ele-ments.


Definition. An element a ∈ A of a bialgebra A is said to be primitive if

δ(a) = 1⊗ a+ a⊗ 1.

The set of all primitive elements forms a vector subspace P(A) calledthe primitive subspace of the bialgebra A. The primitive subspace is closedunder the commutator [a, b] = ab − ba, so it forms a Lie algebra (which isAbelian, if A is commutative). Indeed, if a, b ∈ P(A), then

δ(a) = 1⊗ a+ a⊗ 1,

δ(b) = 1⊗ b+ b⊗ 1,

whence

δ(ab) = 1⊗ ab+ a⊗ b+ b⊗ a+ ab⊗ 1,

δ(ba) = 1⊗ ba+ b⊗ a+ a⊗ b+ ba⊗ 1

and therefore

δ([a, b]) = 1⊗ [a, b] + [a, b]⊗ 1.

Definition. An element a ∈ A is said to be semigroup-like if and

δ(a) = a⊗ a.If a is, moreover, invertible, then it is called group-like.

The set of all group-like elements G(A) of a bialgebraA is a multiplicativegroup. Indeed, G(A) consists of all invertible elements of the semigroupa ∈ A | δ(a) = a⊗ a.

Among the examples of bialgebras given above, the notions of the prim-itive and group-like elements are especially transparent in the case A = FG

(Example 2). As follows from the definitions, primitive elements are additivefunctions (f(xy) = f(x) + f(y)) while group-like elements are multiplicativefunctions (f(xy) = f(x)f(y)).

In Example 3, using the isomorphism S(X)⊗S(X) ∼= S(X⊕X), we canrewrite the definition of the coproduct as δ(x) = (x, x) ∈ X ⊕X for x ∈ X.It is even more suggestive to view the elements of the symmetric algebraA = S(X) as polynomial functions on the dual space X∗ (where homoge-neous subspaces S0(X), S1(X), S2(X), etc. correspond to constants, linearfunctions, quadratic functions etc. on X∗). In these terms, the product in Acorresponds to the usual (pointwise) multiplication of functions, while thecoproduct δ : S(X)→ S(X ⊕X) acts according to the rule

δ(f)(ξ, η) = f(ξ + η), ξ, η ∈ X∗.Under the same identifications,

(f ⊗ g)(ξ, η) = f(ξ)g(η),

340 Appendix

in particular,

(f ⊗ 1)(ξ, η) = f(ξ),

(1⊗ f)(ξ, η) = f(η).

We see that an element of S(X), cosidered as a function on X∗, is primitive(group-like) if and only if this function is additive (multiplicative):

f(ξ, η) = f(ξ) + f(η),

f(ξ, η) = f(ξ)f(η).

The first condition means that f is a linear function on X∗, i.e. it corre-sponds to an element of X itself; therefore,

P(S(X)) = X.

Over a field of characteristic zero, the second condition cannot hold forpolynomial functions except for the constant function equal to 1; thus

G(S(X)) = 1.The completed symmetric algebra S(X), in contrast with S(X), has a lot ofgroup-like elements. As shows Quillen’s theorem (see page 347),

G(S(X)) = exp(x) | x ∈ X,where exp(x) is defined as a formal power series 1 + x+ x2/2! + . . .

A.2.7. Exercise. Describe the primitive and group-like elements in exam-ples 1 and 4.

(Answer to 1: P = 0, G = G.

Answer to 4: 4P = g, G = 0 (in the completed case G = exp(g).)

A.2.8. Dual bialgebras. The dual vector space of a bialgebra can be nat-urally equipped with a structure of a bialgebra, too.

Let A be a bialgebra over a field F and let A∗ = HomF(A,F). We definethe product of two elements f, g ∈ A∗ as

(1) (f · g)(a) = (f ⊗ g)(δ(a)) =∑

i

f(a′i)g(a′′i ),

if δ(a) =∑

i a′i ⊗ a′′i .

The coproduct of an element f ∈ A∗ is defined by the equation

(2) δ(f)(a⊗ b) = f(ab).

where the evaluation of δ(f) ∈ A∗⊗A∗ on the element a⊗ b ∈ A⊗A comesfrom the natural pairing between A∗ ⊗A∗ and A⊗A.

Thus, the product (resp. coproduct) in A∗ is defined through the co-product (resp. product) in A. It is easy to define the unit and the counit in


A∗ in terms of the counit and the unit of A and then check that the spaceA∗ becomes indeed a bialgebra.

If read in the opposite direction, Equations 1 and 2 allow one to uniquelydefine the structure of a bialgebra in the initial space A, if such a structureis given in the dual space A∗. The uniqueness follows from the simpleobservation that an element a of a vector space A is completely defined bythe values of all linear functionals f(a), while an element of the tensor squareq ∈ A⊗A is completely determined by values (f ⊗ g)(q) for all f, j ∈ A∗.A.2.9. Exercise. Check that the axioms of a bialgebra in the dual spaceA∗ imply that A becomes a bialgebra, too.

A.2.10. Exercise. Check that the bialgebra of Example 2 above is dual tothe bialgebra of Example 1.

A.2.11. Exercise. Prove that for a finite-dimensional bialgebra A there isa natural isomorphism (A∗)∗ ∼= A.

If A is infinite-dimensional, then the assertion of the last Exercise doesnot hold: in this case the vector space (A∗)∗ is strictly bigger than A.

A.2.12. Proposition. Primitive (resp. group-like) elements in the bialge-bra A∗ are linear functions on A which are additive (resp. multiplicative),i.e. satisfy the respective identities

f(ab) = f(a) + f(b),

f(ab) = f(a)f(b)

for all a, b ∈ A.

Proof. An element f ∈ A∗ is primitive if δ(f) = 1⊗ f + f ⊗ 1. Evaluatingthis on an arbitrary tensor product a⊗ b with a, b ∈ A, we obtain

f(ab) = f(a) + f(b).

An element f ∈ A∗ is group-like if δ(f) = f ⊗ f . Evaluating this on anarbitrary tensor product a⊗ b, we obtain

f(ab) = f(a)f(b).

A.2.13. Filtrations and gradings in vector spaces. A decreasing fil-tration (d-filtration) in a vector space A over a field F is a sequence ofsubspaces Ai, i = 0, 1, 2, ... such that

A = A0 ⊃ A1 ⊃ A2 ⊃ . . .

342 Appendix

The order of an element a ∈ A is defined as the natural number n such thata ∈ An, but a 6∈ An−1. If no such number exists, we set ord(a) = ∞. Thefactors of a d-filtration are the quotient spaces GiA = Ai/Ai+1.

An increasing filtration (i-filtration) in a vector space A is a sequence ofsubspaces Ai, i = 0, 1, 2, ... such that

A0 ⊂ A1 ⊂ A2 ⊂ · · · ⊂ A.The order of an element a ∈ A is defined as the natural number n such thata ∈ An, but a 6∈ An+1. If no such number exists, we set ord(a) = ∞. Thefactors of an i-filtration are the quotient spaces GiA = Ai/Ai−1, where bydefinition A−1 = 0.

A filtration (either d- or i-) is said to be of finite type if all its factors arefinite-dimensional. Note that in each case the whole space has a (possiblyinfinite-dimensional) ‘part’ not covered by the factors, viz. ∩∞i=1Ai for ad-filtration and A/∪∞i=1 Ai for an i-filtration. If these parts vanish, then wesay that we deal with a reduced filtered space.

There is a simple procedure way to obtain a reduced filtered space withthe same factors:

• A′ = A/ ∩∞i=1 Ai for a d-filtered space,

• A′ = ∪∞i=1Ai for an i-filtered space.

In either case the reduced space inherits the filtration in a natural way.

A filtered basis of an i-filtered vector space A is constructed as follows.One takes a basis of A0, then adds some vectors to form a basis of A1 ⊃ A0

etc. For a d-filtered space A, a filtered basis is defined as the disjoint unionof a countable number of subsets T = T0 ∪ T1 ∪ T2 ∪ · · · ∪ T ′ where T ′ isa basis of ∩∞i=1Ai (possibly infinite), while all Ti’s are finite sets with theproperty that for each n the union of ∪∞i=nTi and T ′ is a basis of An.

A vector space is said to be graded if it is represented as a direct sum ofits subspaces

A =∞⊕

i=0

Ai.

With a filtered vector space A, one can associate a graded space G(A) setting

G(A) =∞∑

i=0

GiA =∞∑

i=0

Ai/Ai+1

in case of a d-filtration and

G(A) =

∞∑

i=0

GiA =

∞∑

i=0

Ai/Ai−1

in case of an i-filtration.


If A is a filtered space of finite type, then the homogeneous componentsGiA are also finite-dimensional; the ‘size’ of G(A) has a compact descriptionby means of the Poincare series

∑∞k=0 dim(GiA)tk, where t is an auxiliary

formal variable.

A.2.14. Example. The Poincare series of the algebra of polynomials inone variable is

1 + t+ t2 + ... =1

1− t .

A.2.15. Exercise. Find the Poincare series of the polynomial algebra withn independent variables.

Add a formal definiton of the graded completion!!

A.2.16. Dual filtration. Given a d-filtered vector space A (i.e. endowedwith a decreasing filtration) one can introduce a natural structure of ani-filtered space in the dual space A∗ = Hom(A,F) by the formula

(A∗)i = f ∈ A∗ : f |Ai+1 = 0.Note that the subscript of A is i + 1, not i. This convention is crucial toensure the following important assertion.

A.2.17. Lemma. Each factor of the dual filtered space is dual to the factorof the initial space with the same number: Gi(A

∗) = (GiA)∗. Hence, for ani-filtered space of finite type, we have an (unnatural) isomorphism Gi(A

∗) ∼=GiA.

Proof. Indeed, there is a natural map λ : (A∗)i → (GiA)∗ defined by theformula λ(f) = f |Ai . Since any linear function can be extended from asubspace to the whole space, this mapping is surjective. Its kernel is bydefinition (A∗)i−1, whence the assertion.

A.2.18. Filtered bialgebras.

Definition. We will say that a bialgebra A is d-filtered (resp. i-filtered)if its underlying vector space has an decreasing (resp. increasing) filtrationby subspaces Ai compatible with the algebraic operations in the followingsense:

ApAq ⊂ Ap+q for p, q ≥ 0, δ(An) ⊂∑

p+q=n

Ap ⊗Aq for n ≥ 0.

The second condition can be equivalently rewritten as δ(An) ⊂∑

p+q≥nAp⊗Aq for i-filtered bialgebras, and as δ(An) ⊂

∑p+q≤nAp ⊗ Aq for d-filtered

bialgebras.

Additional requirements imposed on the unit and the counit are: 1 ∈ A0

and ε|A1 = 0.

344 Appendix

A.2.19. Dual filtered bialgebra.

Theorem. The bialgebra dual to a d-filtered bialgebra is an i-filtered bial-gebra.

Proof. Let A be a bialgebra with a decreasing filtration by subspaces Ai.The constructions of Sec. A.2.8 and A.2.16 define a structure of a bialgebraand shat of a filtered vector space in the dual space A∗. We must check thatthese two structures are compatible, i.e. that

(A∗)p(A∗)q ⊂ (A∗)p+q

andδ((A∗)n) ⊂

∑

p+q=n

(A∗)p ⊗ (A∗)q .

To prove the first assertion, let f ∈ (A∗)p, g ∈ (A∗)q and a ∈ Ap+q+1.Write δ(a) as a sum of tensor products a′ ⊗ a′′ where ord(a′) + ord(a′′) ≥p+ q + 1. Then by definition

(fg)(a) = (f ⊗ g)(δ(a)) =∑

f(a′)g(a′′) = 0,

because in each summand either ord(a′) ≥ p + 1 and then f(a′) = 0 orord(a′′) ≥ q + 1 and then g(a′′) = 0.

For the second assertion, pick an f ∈ (A∗)n. Choose a filtered basis (p.342) in A and dual filtered basis in A∗ and expand δ(f) over the correspond-ing basis of the tensor square A∗ ⊗ A∗. Let λf ′ ⊗ f ′′ be one of the termsof this expansion and let a′, a′′ ∈ A be the vectors of the dual basis of Aconjugate to f ′ and f ′′. Denote b = a′a′′. Then

f(b) = δ(f)(a′ ⊗ a′′) = λ 6= 0.

According to the definition of the dual filtration, this inequality implies thatord(b) ≤ ord(f) = n. Note, finally, that ord(a′) + ord(a′′) ≤ ord(a′a′′) in ad-filtered algebra.

A.2.20. Hopf algebras.

A.2.21. Definition. A Hopf algebra is a graded bialgebra. This meansthat the vector space A is graded by integer numbered subspaces

A =⊕

k≥0

Ak

and the grading is compatible with the operations µ, ι, δ, ε in the followingsense:

µ : Am ⊗An → Am+n,

δ : An →⊕

k+l=n

Ak ⊗Al.


A Hopf algebra A is said to be of finite type, if all its homogeneous com-ponents An are finite-dimensional. A Hopf algebra is said to be connected,if (1) ι : F → A is an isomorphism of F onto A0 ⊂ A and (2) ε |Ak

= 0 fork > 0, while ε |A0 is an isomorphism between A0 and F, inverse to ι.

Remark 1. The above definition follows the classical paper [MiMo].In more recent times, it became customary to include one more operation,called antipode, in the definition of a Hopf algebra. The antipode is a linearmapping S : A→ A such that µ (S ⊗ 1) δ = µ (1⊗ S) δ = ι ε. Notethat the bialgebras we will be mostly interested in (those that satisfy thepremises of Theorem A.2.24 below) always have an antipode.

Remark 2. We will need to extend the definition of a Hopf algebra fromgraded algebras, i.e. direct sums of finite-dimensional homogeneous compo-nents, to completed graded algebras, i.e. direct products of such components(see p. 96).

A.2.22. Exercise. If A is a filtered bialgebra, then its associated gradedvector space G(A) carries the structure of a Hopf algebra with all algebraicoperations naturally defined by the operations in A by passing to quotientspaces.

A.2.23. Dual Hopf algebra. Here we specialize the construction of Sec-tion A.2.8 to the case of Hopf algebras.

If A = ⊕Ak is a Hopf algebra of finite type and Wk = Hom(Ak,F)are vector spaces, dual to the homogeneous components of A, then the dualspace to the whole A is represented as the product

∏kWk which is in general

bigger than the direct sum W =∑

kWk. However, it is this smaller spaceW that we will call the dual Hopf algebra of A with the operations inducedby the corresponding operations in A as follows:

µ∗ : Wn → ⊕k+l=n

Hom(Ak ⊗Al,F) ∼= ⊕k+l=n

Wk ⊗Wl is a comultiplication for W

δ∗ : Wn ⊗Wm →Wm+n is a multiplication for W

ι∗ : W → F is a counit for W

ε∗ : F→W is a unit for W

A.2.24. Structure theorem for Hopf algebras. The bialgebra of chorddiagrams A, which we will construct in 4.5, has the following properties: itis commutative, cocommutative, connected, and of finite type. (We do notlist the properties of associativity and coassociativity which are presupposedfor every bialgebra).

346 Appendix

It turns out that these properties ensure that A has a straightforwarddescription in terms of its primitive elements (page 339).

Is it easy to see that for a Hopf algebra the primitive subspace P =P (A) ⊂ A is homogeneous in the sense that P =

⊕n≥0

P ∩An.

Theorem. (Milnor–Moore [MiMo]). Any commutative cocommutativeconnected Hopf algebra of finite type over a field of characteristic 0 is equalto the symmetric algebra of its primitive subspace:

A = S(P (A)).

The word ’equal’ (not just isomorphic) means that if the inclusion P (A)→ Ais extended to a homomorphism S(P (A)) → A in the standard way, thisgives a bialgebra isomorphism. In other words, if a linear basis is chosen inevery homogeneous component Pn = P ∩ An, then every element of A canbe written uniquely as a polynomial in these variables.

Proof. There are two assertions to prove:

(I) that every element of A is polynomially expressible (i. e. as a linearcombination of products) through primitive elements,

(II) that the value of a nonzero polynomial on a set of linearly indepen-dent homogeneous primitive elements cannot vanish in A.

Let us start to prove assertion (I) for homogeneous elements of the al-gebra by induction on their degree.

First note that under our assumptions the coproduct of a homogeneouselement a ∈ An has the form

(3) δ(x) = 1⊗ x+ · · ·+ x⊗ 1,

where the dots stand for an element of A1⊗An−1 + · · ·+An−1⊗A1. Indeed,we can always write δ(x) = 1⊗ y + · · ·+ z ⊗ 1. By cocommutativity y = z.Then, x = (ε⊗ id)(δ(x)) = y + 0 + · · ·+ 0 = y.

In particular, for any element x ∈ A1 equation (3) ensures that δ(x) =1⊗ x+ x⊗ 1, so that A1 = P1. (It may happen that A1 = 0, but this doesnot interfere the subsequent argument!)

Take an element x ∈ A2. We have

δ(x) = 1⊗ x+∑

λijp1i ⊗ p1

j + x⊗ 1,

where p1i constitute a basis of A1 = P1 and λij is a symmetric matrix over

the ground field. Let

x′ =1

2

∑λijp

1i p

1j .

Thenδ(x′) = 1⊗ x′ +

∑λijp

1i ⊗ p1

j + x′ ⊗ 1.


It follows that

δ(x− x′) = 1⊗ x′ + x′ ⊗ 1,

i. e. x − x′ is primitive, and x is expressed through primitive elements as(x− x′) + x′, which is a polynomial, linear in P2 and quadratic in P1.

Proceeding in this way, assertion I can be proved in grading 3, 4, ... Weomit the formal inductive argument.

The uniqueness of the polynomial representation in terms of basic prim-itive elements (assertion II) is a consequence of the following observation:

Denote by Rn ⊂ An the subspace spanned by all nontrivial productsof homogeneous elements of positive degree. Then there is a direct sumdecomposition

An = Pn ⊕Rn.This completes the proof.

!!! correct the last ugly argument!!!

A.2.25. Corollary. An algebra A satisfying the assumptions of the theorem

1. has no zero divisors,

2. has the antipode S defined on primitive elements (see p. 339) by

S(p) = −p .!!! Add the structure theorem for Hopf algebras that arenot commutative, but cocommutative (they are univer-sal envelops of their primitive spaces). Same corollarieshold. Also, add an example of a non-cocommutative alge-bra where the theorem does not hold at all!!!

A.2.26. Group-like elements in Hopf algebras.

A.2.27. Lemma. (D. Quillen’s theorem [Q]) For a completed connectedHopf algebra over a field of characteristic 0 the functions exp and log (definedby the usual Taylor expansions) establish one-to-one mappings between theset of primitive elements P (A) and the set of group-like elements G(A).

Proof. Let p ∈ P (A). Then

δ(pn) = (1⊗ p+ p⊗ 1)n =∑

k+l=n

n!

k!l!pk ⊗ pl

and therefore

δ(ep) = δ(

∞∑

n=0

pn

n!) =

∞∑

k=0

∞∑

l=0

1

k!l!pk ⊗ pl =

∞∑

k=0

1

k!pk ⊗

∞∑

l=0

1

l!pl = δ(ep)⊗ δ(ep)

348 Appendix

which means that ep ∈ G(A).

Vice versa, assuming that g ∈ G(A) we want to prove that log(g) ∈P (A). By assumption, our Hopf algebra is connected which implies thatthe graded component g0 ∈ A0

∼= F is equal to 1. Therefore we can writeg = 1+h where h ∈∑k>0Ak. The condition that g is group-like transcribesas

(4) δ(h) = 1⊗ h+ h⊗ 1 + h⊗ h.Now,

p = log(g) = log(1 + h) =

∞∑

k=1

(−1)k−1

khk

and an exercise in power series combinatorics shows that equation 4 impliesthe required property

δ(p) = 1⊗ p+ p⊗ 1.

A.3. Free associative and free Lie algebras

This section contains a compendium of results on free associative and freeLie algebras important for the study of the Drinfeld associator (Chapter 12).We give almost no proofs referring the interested reader to special literature,e.g. [Reu].

A.3.1. Definition. The free (associative) algebra with n generators F(n)over a field F is the algebra of non-commutative polynomials in n variables.

For example, the algebra F(2) = F〈x1, x2〉 consists of finite linear com-binations of the form c+c1x1+c2x2+c11x

21+c12x1x2+c21x2x1+c22x

22+ . . . ,

cα ∈ F, with natural addition and multiplication.

The meaning of the word free is that, in F(n), there are no relationsbetween the generators and thus the only identities that hold in F(n) arethose that follow from the axioms of an algebra, for example, (x1 + x2)

2 =x2

1 + x1x2 + x2x1 + x22. An abstract way to define the free algebra uniquely

up to isomorphism, is by means of the universal property:

if A is an arbitrary (associative, not necessarily commuta-tive) algebra and y1, . . . , yn ∈ A a set of its elements, thenthere is a unique homomorphism F(n) → A that takeseach xi into the corresponding yi.

We are interested in the relation between associative algebras and Liealgebras, and we begin with the simple observation that any associativealgebra has the structure of a Lie algebra defined by the conventional rule[a, b] = ab− ba.

A.3. Free associative and free Lie algebras 349

Definition. A commutator monomial of degree m and depth k in a freealgebra F is a monomial in the generators xi1 · · · · · xim with k pairs ofbrackets inserted in arbitrary position.

For example, x2[x1, [x1, x23]] is a commutator monomial of degree 5 and

depth 2. The maximal depth of a commutator monomial of degree m ism − 1, and we will call monomials of maximal depth full commutators. Itis easy to see that the linear span of full commutators is closed under thecommutator, so it forms a Lie subalgebra L(n) ⊂ F(n). Using the skew-symmetry and the Jacobi identity, one can rewrite any full commutator asa linear combination of right-normalized full commutators

[x1 · · · · · xm] := [x1, [...[xm−1, xm]...]] .

The following table shows dimensions and indicates some bases of thehomogeneous components of small degree for the algebra L(2):

m dimL(2)m basis1 2 x, y2 1 [x, y]3 2 [x, [x, y]] [y, [x, y]]4 3 [x, [x, [x, [x, y]]]] [y, [x, [x, [x, y]]]] [y, [y, [x, [x, y]]]]5 6 [x, [x, [x, [x, y]]]] [y, [x, [x, [x, y]]]] [y, [y, [x, [x, y]]]]

[y, [y, [y, [x, y]]]] [[x, y], [x, [x, y]]] [[x, y], [y, [x, y]]]

The free algebra is a graded algebra with the grading defined by theconventional degree of monomials: deg(xi1 · · · · ·xik) = k. The homogeneouscomponent F(n)k of degree k has dimension nk, and the Poincare series ofF(n) is dim(F(n)k)t

k = 1/(1− nt).Let A be an associative algebra or a Lie algebra. The (first) commu-

tant of A, denoted by A′ = [A,A], is, by definition, the linear span of allcommutators [a, b] (in the associative case defined by the conventional rule[a, b] = ab− ba). There are two ways to iterate the operation A 7→ A′:

• A(1) = A′, A(n+1) = [A,A(n)], leading to the decreasing filtrationof A by subspaces A(n) called the lower central series of A,

• A(1) = A′, A(n+1) = [A(n), A(n)], leading to the filtration A ⊃A(1) ⊃ A(2) ⊃ . . . called the derived series of A.

Obviously, A(n) ⊂ A(n) for any n, and this inclusion is in general strictif n > 1 (for example, this is so in the case of free algebras). The second

subspace of the derived series A(2) = [[A,A], [A,A]] has a special name: itis called the second commutant of A.

350 Appendix

In the case of the free associative algebra, the terms of both the derivedseries and the lower central series are homogeneous subspaces with respectto the grading.

A.3.2. Definition. The completed free algebra with n generators Fn isthe graded completion of F(n), i.e. the algebra of formal power series in nnon-commuting variables, denoted also by F〈〈x1, . . . , xn〉〉.

In the free algebra there is a coproduct δ : F(n)→ F(n)⊗F(n) definedon the generators by δ(xi) = xi ⊗ xi and then extended by linearity andmultiplicativity to the entire F(n).

A.3.3. Theorem. The primitive space P(F(n)) is equal to the linear spanof all full commutators.

There is a natural linear operator α : F(n) → F(n) defined on themonomials by

α(x1 · · · · · xk) =1

k![x1 · · · · · xk],

where [x1·· · ··xk] denotes the full right-normalized commutator [x1, [...[xk−1, xk]...]].

A.3.4. Theorem. The operator α is a projector onto the subspace L(n) ⊂F(n).

———————- (moved from chapter 9: brush up!!)

A.3.5. Free associative algebra and free Lie algebra. Let C=C〈〈A,B〉〉be the algebra of formal power series in two non-commuting variables, i.e.the completed free associative algebra with two generators.

Coproduct in C — useful intuition.

If C is understood as the space of functions in two non-commuting vari-ables A and B, then C ⊗ C can be be interpreted as the space of functionsin four variables A1, B1, A2, B2 where each of A1, B1 commutes with eachof A2, B2, but variables with the same subscript do not commute betweenthemselves. In this setting, the coproduct has the following transparentmeaning:

δ(ϕ(A,B)) = ϕ(A1 +A2, B1 +B2).

In particular, the primitive elements of C are precisely the functions havingthe additivity property

ϕ(A1 +A2, B1 +B2) = ϕ(A1, B1) + ϕ(A2, B2).

Bibliography

[AdC] Colin C. Adams, The Knot Book. W. H. Freeman and Company, New York, 1999.

[AdJ] J. Frank Adams, Lectures on Lie Groups. W. A. Benjamin Inc., 1969.

[Adya] S. I. Adyan, Fragments of the word ∆ in the braid group, Math. Notes 36(1-2)(1984) 505–510.

[Al] J. W. Alexander, Topological invariants of knots and links, Trans. AMS 30 (1928)275–306.

[Al1] J. W. Alexander, A lemma on system of knotted curves, Proc. Nat. Acad. Sci.USA 9 (1923) 93–95.

[Ar1] V. I. Arnold, Cohomology ring of the group of pure braids, Matem. Zametki 5

(2), 227–231 (1969).

[Ar2] V. I. Arnold, Vassiliev’s theory of discriminants and knots, First European Con-gress of Mathematicians, (Paris, July 1992), Birkhauser, Basel–Boston–Berlin,1, 3–29 (1994).

[Ar3] V. I. Arnold, Topological Invariants of Plane Curves and Caustics, UniversityLecture Series 5, American Mathematical Society, 1994.

[At] M. F. Atiyah, The Geometry and Physics of Knots, Cambridge University Press,1990.

[Big1] S. Bigelow, The Burau representation of the braid group Bn is not faith-

ful for n = 5, Geometry and Topology 3 (1999) 397–404 (also available atarXiv:math.GT/9904100).

[Big2] S. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14(2) (2001) 471–486(also available at arXiv:math.GT/0005038).

[Bir1] J. S. Birman, Braids, Links and Mapping Class Groups, Princeton UniversityPress, 1974.

[Bir2] J. S. Birman, New points of view in knot theory, Bull. Amer. Math. Soc. 28

(1993) 253–287.

[BL] J. S. Birman and X.-S. Lin, Knot polynomials and Vassiliev’s invariants, Invent.Math. 111 (1993) 225–270.

[BN0] D. Bar-Natan, Weights of Feynman diagrams and the Vassiliev knot invariants,preprint (February 1991), http://www.math.toronto.edu/~drorbn/papers.

351

352 Bibliography

[BLT] D. Bar-Natan, T. Q. T. Le, D. P. Thurston, Two applications of elementary knot

theory to Lie algebras and Vassiliev invariants, Geometry and Topology 7(1)(2003) 1-31, arXiv:math.QA/0204311.

[BNt] D. Bar-Natan, Perturbative Aspects of the Chern-Simons Topological

Quantum Field Theory, Ph.D. thesis, Harvard University (June 1991).http://www.math.toronto.edu/~drorbn/papers.

[BN1] D. Bar-Natan, On the Vassiliev knot invariants, Topology, 34 (1995) 423–472.

[BN1a] D. Bar-Natan, On the Vassiliev knot invariants, Draft version of the previouspaper (edition: August 21, 1992).

[BN2] D. Bar-Natan, Non-associative tangles, in Geometric topology (proceedings ofthe Georgia international topology conference), (W. H. Kazez, ed.), 139–183,Amer. Math. Soc. and International Press, Providence, 1997.

[BN3] D. Bar-Natan, Lie algebras and the four color theorem, Combinatorica 17-1 (1997)43–52 (also available at arXiv:math.QA/9606016).

[BN4] D. Bar-Natan, Vassiliev Homotopy String Link Invariants, Journal of Knot The-ory and its Ramifications 4-1 (1995) 13–32.

[BN5] D. Bar-Natan, Computer programs and data files, available on-line fromhttp://www.math.toronto.edu/~drorbn/papers/nat/natmath.html.

[BN6] D. Bar-Natan, On associators and the Grothendieck-Teichmuller group. I, SelectaMath. (N.S.), 4 (1998) 183–212.

[BN7] D. Bar-Natan, Some Computations Related to Vassiliev Invariants, (last updatedMay 5, 1996), http://www.math.toronto.edu/~drorbn/papers.

[BGRT] D. Bar-Natan, S. Garoufalidis, L. Rozansky, D. Thurston. Wheels, wheeling, and

the Kontsevich integral of the unknot. Israel Journal of Mathematics 119 (2000)217-237, arXiv:q-alg/9703025.

[BNG] D. Bar-Natan, S. Garoufalidis, On the Melvin–Morton–Rozansky conjecture, In-vent. Math., 125 (1996) 103–133.

[BNL] D. Bar-Natan and R. Lawrence, A Rational Surgery Formula for the LMO Invari-ant, Israel Journal of Mathematics 140 (2004) 29–60, arXiv:math.GT/0007045.

[BBBL] J. Borwein, D. Bradley, D. Broadhurst, P. Lisonek, Combinatorial aspects of

multiple zeta values, Electron. J. Combin. 5(1) (1998) #R38 (see also preprintmath.NT/9812020).

[BR1] B. Bollobas, O. Riordan, Linearized chord diagrams and an upper bound for

Vassiliev invariants, Journal of Knot Theory and its Ramifications, 9(7) (2000)847–853.

[BR2] B. Bollobas, O. Riordan, A polynomial of graphs on orientable surfaces,Proc. London Math. Soc. 83 (2001), 513–531.

[BR3] B. Bollobas, O. Riordan, A polynomial of graphs on surfaces, Math. Ann. 323(1)(2002), 81–96.

[Br] D. Broadhurst, Conjectured enumeration of Vassiliev invariants, preprintq-alg/9709031.

[BZ] G. Burde, H. Zieschang. Knots. Second edition. Walter de Gruyter, Berlin – NewYork, 2003.

[CE] G. Cairns, D. M. Elton, The planarity problem for signed Gauss words, Journalof Knot Theory and its Ramifications, 2 (1993) 359–367.

[Car1] P. Cartier, Construction combinatoire des invariants de Vassiliev–Kontsevich des

nœuds, C. R. Acad. Sci. Paris 316 Serie I (1993) 1205–1210.

Bibliography 353

[Car2] P. Cartier, Values of the zeta function, Notes of survey lectures for theundegraduate students, Independent University of Moscow, May (1999),http://www.mccme.ru/ium/stcht.html (in Russian).

[Chen] K.-T. Chen, Iterated path integrals, Bull. AMS 83 (1977) 831–879.

[Ch1] S. Chmutov, A proof of Melvin–Morton conjecture and Feynman diagrams, Jour-nal of Knot Theory and its Ramifications, 7 (1998) 23–40.

[Ch2] S. Chmutov, Combinatorial analog of the Melvin–Morton conjecture, Proceedingsof KNOTS ’96, World Scientific Publishing Co. (1997) 257–266.

[CD1] S. Chmutov, S. Duzhin. An upper bound for the number of Vassiliev knot invari-

ants, Journal of Knot Theory and its Ramifications. 3 (1994), 141–151.

[CD2] S. Chmutov, S. Duzhin. A lower bound for the number of Vassiliev knot invari-

ants. Topology and its Applications 92 (1999) 201–223.

[CD3] S. Chmutov, S. Duzhin. The Kontsevich integral. Acta Applicandae Math. 66

(2001), 155–190.

[CD4] S. Duzhin. S. Chmutov. Knots and their invariants. Matematicheskoe Prosvesche-nie, no. 3, 1999, pp. 59–93 (in Russian).

[CDL1] S. Chmutov, S. Duzhin, S. Lando. Vassiliev knot invariants I. Introduction, Adv.in Soviet Math., 21 (1994) 117–126.

[CDL2] S. Chmutov, S. Duzhin, S. Lando, Vassiliev knot invariants II. Intersection graph

conjecture for trees, Advances in Soviet Math., 21 (1994) 127–134.

[CDL3] S. Chmutov, S. Duzhin, S. Lando, Vassiliev knot invariants III. Forest algebra

and weighted graphs, Advances in Soviet Math., 21 (1994) 135–145.

[Con] J. H. Conway, An enumeration of knots and links and some of their algebraic

properties, Computational Problems in Abstract Algebra, Pergamon Press, N.-Y., 1970, 329–358.

[CrF] R. Crowell, R. Fox, Introduction to Knot Theory. Ginn and Co., 1963.

[CV] S. Chmutov, A. Varchenko, Remarks on the Vassiliev knot invariants coming

from sl2, Topology 36 (1997) 153–178.

[CT] J. Conant, P. Teicher, Grope cobordism of classical knots, preprintarXiv:math.GT/0012118.

[Cro] Peter Cromwell, Knots and Links, Cambridge University Press, 2004.

[Da0] O. Dasbach, Private communication, July 1997.

[Da1] O. Dasbach. On subspaces of the space of Vassiliev invariants, Ph.D. thesis.Dusseldorf, August 1996.

[Da2] O. Dasbach. On the combinatorial structure of primitive Vassiliev invariants

II, J. Comb. Theory, Ser. A, Vol. 81, No. 2, 1998, pp. 127–139. (On-line athttp://www.math.uni-duesseldorf.de/~kasten/).

[Da3] O. Dasbach. On the Combinatorial Structure of Primitive Vassiliev Invariants

III — A Lower Bound, Communications in Contemporary Mathematics, Vol. 2,No. 4, 2000, pp. 579–590. Also arXiv:math.GT/9806086.

[Dr1] V. G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz, 1, no. 6, 114–148 (1989).English translation: Leningrad Math. J., 1 (1990) 1419–1457.

[Dr2] V. G. Drinfeld, On quasi-triangular quasi-Hopf algebras and a group closely con-

nected with Gal(Q/Q), Algebra i Analiz, 2, no. 4, 149–181 (1990). English trans-lation: Leningrad Math. J., 2 (1991) 829–860.

[Dr3] V. G. Drinfeld, Quantum Groups, in: Proceedings of the International Congressof Mathematicians, Berkeley, 1989, 798–820.

354 Bibliography

[Dr4] V. G. Drinfeld, Hopf algebras and the quantum Yang–Baxeter equation, Sov.Math. Dokl. 32 (1985) 254–258.

[Du1] S. Duzhin, A quadratic lower bound for the number of primitive Vassiliev in-

variants, Extended abstract, KNOT’96 Conference/Workshop report, WasedaUniversity, Tokyo, July 1996, p. 52–54.

[Du2] S. Duzhin, Lectures on Vassiliev Knot Invariants, “Lectures in MathematicalSciences”, vol. 19, The University of Tokyo, 2002. 123 pp.

[DKC] S. Duzhin, A. Kaishev, S. Chmutov, The algebra of 3-graphs, Proc. SteklovInst. Math. 221(2)(1998) 157–186.

[ElMo] E. El-Rifai, H. Morton, Algorithms for positive braids, Quart. J. Math. Oxford45(180) (1994) 479–497.

[EFK] P. Etingof, I. Frenkel, A. Kirillov Jr., Lectures on representation theory and

Knizhnik—Zamolodchikov equations, Mathematical Surveys and Monographs,58, AMS, Providence, RI, 1998.

[Eu] L. Euler, Meditationes circa singulare serierum genus, Novi commentarii acad-emiae scientiarum Petropolitanae 20 (1775), p. 140–186. (Reprinted in: Opera

Omnia, Ser.1, XV, Teubner, Leipzig–Berlin (1927), p. 217–267).

[Fi] Th. Fiedler, Gauss diagram invariants for knots and links, Dordrecht, Boston:Kluwer Academic Publishers, 2001.

[FKV] J. M. Figueroa-O’Farrill, T. Kimura, A. Vaintrob, The universal Vassiliev in-

variant for the Lie superalgebra gl(1|1), Comm. Math. Phys. 185 (1997), no. 1,93–127. (Draft version available online at arXiv:math.QA/9602014).

[FrYe] P. Freed, D. Yetter, Braided compact closed categories with applications to low

dimensional topology, Adv. in Math. 77(2) (1989) 156–182.

[FH] W. Fulton, J. Harris, Representation Theory, A first course, Graduate Texts inMathematics 129, Springer-Verlag, 1991.

[Gar] F. Garside, The braid group and other groups, Quart. J. Math. Oxford 20(78)(1969) 235–254.

[GK] S. Garoufalidis and A. Kricker, A rational noncommutative invariant of boundarylinks, Geometry and Topology 8-4 (2004) 115–204, arXiv:math.GT/0105028.

[G1] M. Goussarov, A new form of the Conway-Jones polynomial of oriented links,Zapiski nauch. sem. POMI 193 (1991), 4–9 (English translation: Topology ofmanifolds and varieties (O. Viro, editor), Amer. Math. Soc., Providence 1994,167–172).

[G2] M. Goussarov, On n-equivalence of knots and invariants of finite degree, Za-piski nauch. sem. POMI 208 (1993), 152–173 (English translation: Topology ofmanifolds and varieties (O. Viro, editor), Amer. Math. Soc., Providence 1994,173–192).

[G3] M. Goussarov, Finite type invariants are presented by Gauss diagram formulas,http://www.ma.huji.ac.il/~drorbn/Goussarov/, preprint, December 1998.

[G4] M. Goussarov, Variations of knotted graphs, geometric technique of n-equivalence,St. Petersburg Math. J., 12, no. 4 (2001) 569–604.

[GPV] M. Goussarov, M. Polyak, O. Viro, Finite type invariants of classical and virtual

knots, Topology, 39 (2000) 1045–1068.

[Gon] A. Goncharov. Multiple polylogarithms, cyclotomy and modular complexes, Math.Res. Lett. 5 (1998) 497–516.

Bibliography 355

[Gor1] V. Goryunov, Vassiliev invariants of knots in R3 and in a solid torus,American Mathematical Society Translations (2), 190 (1999) 37–59(http://www.liv.ac.uk/esu14/knotprints.html).

[Gor2] V. Goryunov, Vassiliev type invariants in Arnold’s J+-theory of plane

curves without direct self-tangencies, Topology 37 (1998) 603–620(http://www.liv.ac.uk/esu14/knotprints.html).

[Ha1] K. Habiro, Aru musubime no kyokusyo sousa no zoku ni tuite, Master thesis atTokyo University (in Japanese), 1994.

[Ha2] K. Habiro, Claspers and finite type invariants of links, Geom. Topol., 4 (2000)1–83. http://www.maths.warwick.ac.uk/gt/GTVol4/paper1.abs.html

[Hoff] M. Hoffman, Multiple harmonic series, Pacific J. Math. 152 (1992) 275–290.

[HoOh] M. Hoffman, Y. Ohno, Relations of multiple zeta values and their algebraic ex-

pression, preprint math.QA/0010140.

[HOM] P. Freyd, D. Yetter, J. Host, W. B. R. Lickorish, K. Millett, A. Ocneanu, A new

polynomial invariant of knots and links, Bull. AMS 12 (1985) 239-246.

[HTW] J. Hoste, M. Thistlethwaite and J. Weeks, The first 1,701,936 knots. Mathemat-ical Intelligencer, 20 (1998) 33–48.

[Hum] J. Humphreys, Introduction to Lie Algebras and Representation Theory, Gradu-ate Texts in Mathematics 9, Springer-Verlag, 1994.

[Jimb] M. Jimbo, A q-difference analogue od U(g) and the Yang–Baxter equation, Lett.Math. Phys. 10 (1985) 63–69.

[Jan] J. Jantzen, Lectures on quantum groups, Graduate Studies in Math. 6, AMS(1996).

[Jo1] V. Jones, A polynomial invariant for knots via von Neumann algebras, Bull.Amer. Math. Soc. (New Series) 12 (1985) 103–111.

[Jo2] V. Jones, Hecke algebra representations of braid groups and link polynomials,Math. Ann. 126 (1987) 335–388.

[Jo3] V. Jones, On knot invariants related to some statistical mechanics models, PacificJ. Math. 137 (1989) 311–334.

[Kac] V. Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. 53 (1977) 31–64.

[Kai] A. Kaishev, A program system for the study of combinatorial algebraic invari-

ants of topological objects of small dimension, Ph.D. thesis, Program SystemsIsnstitute of the Russian Academy of Sciences, Pereslavl-Zalessky, 2000.

[Kal] E. Kalfagianni, Finite type invariants for knots in three manifolds, Topology 37-3(1998) 673-707.

[Ka1] L. Kauffman, Formal knot theory, Mathematical Notes 30, Princeton UniversityPress (1983).

[Ka2] L. Kauffman, The Arf invariant of classical knots, Contemporary Mathematics,44 (1985) 101–106.

[Ka3] L. Kauffman, On Knots, Annals of Mathematics Studies 115, Princeton Univer-sity Press, 1987.

[Ka4] L. Kauffman, An invariant of regular isotopy, Transactions of the AMS, 318,no. 2, 417–471, 1990.

[Ka5] L. Kauffman, Virtual knot theory, European Journal of Combinatorics, 20 (1999)663–690.

[Ka6] L. Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395–407.

356 Bibliography

[Kas] C. Kassel, Quantum groups, Graduate Texts in Math. 155, Springer–Verlag, NewYork — Heidelberg — Berlin (1995).

[KRT] C. Kassel, M. Rosso, V. Turaev, Quantum groups and knot invariants, Panoramaset Syntheses, 5, Societe Mathematique de France, Paris (1997).

[Kaw1] A. Kawauchi, The invertibility problem on amphicheiral excellent knots. Proc.Japan Acad. Ser. A Math. Sci. 55 (1979), no. 10, 399–402.

[Kaw2] A. Kawauchi, A survey of knot theory, Birkhauser, 1996.

[Kir] A. N. Kirillov, Dilogarithm identities, Progress of Theor. Phys. Suppl., 118 (1995)61–142 (http://www.math.nagoya-u.ac.jp/~kirillov/publications.html).

[Kn0] J. Kneissler, The number of primitive Vassiliev invariants up to degree twelve,arXiv:math.QA/9706022, June 1997.

[Kn1] J. Kneissler, On spaces of connected graphs. I. Properties of ladders. in Knots in

Hellas ’98, 252–273, World Sci. Publishing, River Edge, NJ, 2000.

[Kn2] J. Kneissler, On spaces of connected graphs. II. Relations in the algebra Λ. J.Knot Theory Ramifications 10 (2001), no. 5, 667–674.

[Kn3] J. Kneissler, On spaces of connected graphs. III. The ladder filtration. J. KnotTheory Ramifications 10 (2001), no. 5, 675–686.

[KnZa] V. Knizhnik, A. Zamolodchikov, Current algebra and the Wess—Zumino model

in two dimensions, Nucl. Phys. B 247 (1984) 83–103.

[KN] Kobayashi, Nomizu. Differential geometry.

[Koh1] T. Kohno, Serie de Poincare-Koszul associee aux groupes de tresses pures, Invent.Math. 82, 57–75 (1985).

[Koh2] T. Kohno, Monodromy representations of braid groups and Yang–Baxter equa-

tions, Ann. Inst. Fourier 37 (1987) 139–160.

[Koh3] T. Kohno, Vassiliev invariants and the de Rham complex on the space of knots,Contemporary Mathematics 179, 123–138 (1994).

[Koh4] T. Kohno, Conformal field theory and topology, Translations of MathematicalMonographs, 210, AMS, Providence, RI, 2002.

[Kon1] M. Kontsevich, Vassiliev’s knot invariants, Adv. in Soviet Math., 16 Part 2(1993) 137–150.

[Kon2] M. Kontsevich, Fax to S. Chmutov, 1997.

[Kram] D.Krammer, Braid groups are linear, Ann. of Math.(2) 155(1) (2002) 131–156.

[Kre] D. Kreimer, Knots and Feynman diagrams, Cambridge Lecture Notes in Physics,13. Cambridge University Press, Cambridge, 2000.

[Kri1] A. Kricker, Alexander–Conway limits of many Vassiliev weight systems, Jour-nal of Knot Theory and its Ramifications 6(5) (1997) 687–714. PreprintarXiv:q-alg/9603011.

[Kri2] A. Kricker, The lines of the Kontsevich integral and Rozansky’s rationality con-

jecture, Preprint arXiv:math.GT/0005284.

[KSA] A.Kricker, B.Spence, I.Aitchison, Cabling the Vassiliev Invariants, Jour.of Knot Theory and its Ramifications 6 (3) (1997), 327–358. See alsoarXiv:q-alg/9511024.

[Kur] V. Kurlin, Explicit description of compressed logarithms of all Drinfeld associa-

tors, preprint math.GT/0408398.

[KuM] B. I. Kurpita, K. Murasugi, A graphical Approach to the Melvin–Morton conjec-

ture, Part I , Topology and its Applications 82 (1998) 297–316.

Bibliography 357

[Lnd] S. Lando, On a Hopf algebra in graph theory, to appear in J. of Comb. Theory,Series B, http://www.pdmi.ras.ru/~arnsem/papers/.

[Lan] J. Lannes, Sur les invariants de Vassiliev de degre inferieur ou egal a 3,L’Enseignement Mathematique, 39 (1993) 295–316.

[Lew] L. Lewin, Polylogarithms and associated functions, Elsevier North Holland, Inc.,New York, Oxford, 1981.

[Le] T. Q. T. Le, Private communication, November, 1994.

[LM1] T. Q. T. Le, J. Murakami, Kontsevich’s integral for the Homfly polynomial and

relations between values of multiple zeta functions, Topology and its Applications62 (1995) 193–206.

[LM2] T. Q. T. Le, J. Murakami, The universal Vassiliev–Kontsevich invariant for

framed oriented links, Compositio Math. 102 (1996) 41–64.

[LM3] T. Q. T. Le, J. Murakami, Representation of the category of tangles by Kontse-

vich’s iterated integral, Comm. Math. Phys., 168 (1995) 535–562.

[LM4] T. Q. T. Le, J. Murakami, The Kontsevich integral for the Kauffman polynomial,Nagoya Mathematical Journal 142 (1996) 39–65.

[LMO] T. Q. T. Le, J. Murakami, T. Ohtsuki, On a universal perturbative invariant of

3-manifolds, Topology 37 (1998), 539–574.

[Les] C. Lescop, Introduction to the Kontsevich integral of framed tangles, CNRSInstitut Fourier preprint, June 1999. (PostScript file available online athttp://www-fourier.ujf-grenoble.fr/~lescop/publi.html).

[Lieb] J. Lieberum, On Vassiliev invariants not coming from semisimple Lie al-

gebras, Jour. of Knot Theory and its Ramifications 8-5 (1999) 659–666,arXiv:math.QA/9706005.

[Lik] W. B. R. Lickorish, An introduction to knot theory, Springer-Verlag New York,Inc. (1997).

[ML] S. MacLane, Categories for the working mathematician, GTM 5 Springer-VerlagNew York, Inc. (1998).

[Mar] Julien Marche. A computation of Kontsevich integral of torus knots.arXiv:math.GT/0404264.

[Mark] A. A. Markov, Uber die freie Aquivalenz geschlossener Zopfe, Recueil Mathema-tique Moscou 1 (1935) 73–78.

[Ma] S. Matveev, Generalized surgeries of three-dimensional manifolds and representa-

tions of homology spheres, Mathematicheskie Zametki, 42 (1987) 268–278; Eng-lish translation: Mathematical Notes, 42 (1987) 651–656.

[Mel1] B. Mellor, The intersection graph conjecture for loop diagrams, Journal of KnotTheory and its Ramifications 9(2) (2000) 187–211.

[Mel2] B. Mellor, A few weight systems arising from intersection graphs, preprint.

[MeMo] P. M. Melvin, H. R. Morton, The coloured Jones function, Comm. Math. Physics169 (1995) 501–520.

[MiMo] J. Milnor, J. Moore, On the Structure of Hopf Algebras, Ann. of Math. (2) 81

(1995), 211–264.

[Mo] H. R. Morton, The coloured Jones function and Alexander polynomial for torus

knots, Math. Proc. Camb. Phil. Soc. 117 (1995) 129–135.

[MC] H. R. Morton, P. R. Cromwell, Distinguishing mutants by knot polynomials, Jour-nal of Knot Theory and its Ramifications 5 (1996) 225–238.

358 Bibliography

[MN] H. Murakami, Y. Nakanishi, On certain moves generating link-homology, Math.Ann., 284 (1989) 75–89.

[Mur] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topol-ogy 26 (1987) 187–194.

[NaS] S. Naik, T. Stanford, A move on diagrams that generates S-equivalence of knots,preprint arXiv:math.GT/9911005.

[Ng] K. Y. Ng, Groups of Ribbon Knots, Topology 37 (1998) 441–458 (See alsoarXiv:math.QA/9502017).

[Oht1] T. Ohtsuki, Quantum invariants. A study of knots, 3-manifolds and their sets,Series on Knots and Everything, 29, World Scientific, 2002

[Oht2] T. Ohtsuki, A cabling formula for the 2-loop polynomial of knots, preprintmath.GT/0310216.

[OU] J. Okuda, K. Ueno, Relations for Multiple Zeta Values and Mellin Transforms of

Multiple Polylogarithms, preprint math.NT/0301277.

[Oll] Olof-Petter Ostlund, Invariants of knot diagrams and relations among Reide-

meister moves, preprint arXiv:math.GT/0005108.

[Per1] K. A. Perko, On the classification of knots, Notices Amer. Math. Soc. 20 (1973),A453–454.

[Per2] K. A. Perko, Letter to the authors, November 15, 2002.

[Piu] S. Piunikhin, Combinatorial expression for universal Vassiliev link invariant,Comm. Math. Phys., 168 (1995) 1–22.

[Pen] R. Penrose, Applications of negative dimensional tensors, In: Combinatorial ma-thematics and its applications (ed. D. J. A. Welsh), New York, London: AcademicPress, 1971.

[Poi] A. Poincare, Sur les groupes d’equations lineaires, Acta Math., 4 (1884) 201–311.

[Po] M. Polyak, On the algebra of arrow diagrams, Let. Math. Phys. 51 (2000) 275–291(see also http://www.math.tau.ac.il/~polyak).

[PS] V. V. Prasolov, A. B. Sossinsky, Knots, Links, Braids and 3-Manifolds. Transla-tions of Mathematical Monographs, 154. American Mathematical Society, Prov-idence (1997).

[PT] J. Przytycki, P. Traczyk, Invariants of linksa of the Conway type, Kobe J. Math.4 (1988) 115–139.

[PV1] M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int.Math. Res. Notes 11 (1994) 445–454.

[PV2] M. Polyak and O. Viro, On the Casson knot invariant, Jour. of Knot Theory andits Ramifications, to appear, arXiv:math.GT/9903158.

[Q] D. Quillen, Rational homotopy theory, Ann. of Math. (2) 90 (1969), 205–295.

[Reu] Reutenauer. Free Lie algebras.

[Roz1] L. Rozansky, A contribution of the trivial connection to the Jones polynomial and

Witten’s invariant of 3d manifolds, Commun. Math. Phys. 175 (1996) 267–296.

[Roz2] L. Rozansky, A rationality conjecture about Kontsevich integral of knots and its

implications to the structure of the colored Jones polynomial , Topology and itsApplications 127 (2003) 47–76. Preprint arXiv:math.GT/0106097.

[Rei] K. Reidemeister, Knot Theory, Chelsea Publ. Co., N.-Y., 1948.

[RT1] N. Reshetikhin, V. Turaev, Ribbon graphs and their invariants derived from quan-

tum groups, Comm. Math. Phys. 127 (1990) 1–26.

Bibliography 359

[RT2] N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and

quantum groups, Invent. Math. 103 (1991), 547–597.

[Rol] D. Rolfsen, Knots and Links. Publish or Perish, 1976.

[Roos] Guy Roos. Functional equations of the dilogarithm and the Kontsevich integral.

Preprint. St.Petersburg, 2003.

[Sos] A. Sossinsky, Knots. Harvard University Press, 2002.

[Spi] M. Spivak, Differential geometry.

[Sta1] T. Stanford, Some computational results on mod 2 finite-type invariants of knots

and string links, In: Invariants of knots and 3-manifolds (Kyoto 2001), pp. 363–376.

[Sta2] T. Stanford, Finite type invariants of knots, links, and graphs, Topology 35, no. 4(1996).

[Sta3] T. Stanford, Vassiliev invariants and knots modulo pure braid subgroups, to ap-pear in Ann. Math. arXiv:math.GT/9907071.

[Sto1] A. Stoimenow, Enumeration of chord diagrams and an upper bound for Vassiliev

invariants, Jour. of Knot Theory and its Ramifications 7(1) (1998) 94–114.

[Sto2] A. Stoimenow, Polynomials of knots with up to 10 crossings, online athttp://www.ms.u-tokyo.ac.jp/~stoimeno.

[Th] M. Thistlethwaite, A spanning tree expansion for the Jones polynomial, Topology26 (1987) 297–309.

[Tro] H.F.Trotter, Non-invertible knots exist. Topology 2 (1964), 275–280.

[Tur1] V. Turaev, A simple proof of the Murasugi and Kauffman theorems on alternating

links, L’Enseignement Mathematique 33 (1987) 203–225.

[Tur2] V. Turaev, The Yang–Baxter equation and invariants of links, Invent. Math. 92

(1988) 527–553.

[Tur3] V. Turaev, Operator invariants of tangles, and R-matrices, Math. USSR Izvestiya35(2) (1990) 411–444.

[TY] K. Taniyama, A. Yasuhara, Band decomposition of knots and Vassiliev invariants,preprint arXiv:math.GT/0003021.

[Tu1] S. D. Tyurina, On formulas of Lannes and Viro-Polyak type for invariants of

finite order, Mathematical Notes, 66 (1999) 525–530.

[Tu2] S. D. Tyurina, Diagrammatic formulae of Viro–Polyak type for knot invariants

of finite order, Russian Mathematical Surveys, 54 (1999) 658–659.

[Vai1] A. Vaintrob, Universal weight systems s and the Melvin–Morton expansion of

the colored Jones knot invariant , J. Math. Sci. 82(1) (1996) 3240–3254. PreprintarXiv:q-alg/9605003.

[Vai2] A. Vaintrob, Melvin–Morton conjecture and primitive Feynman diagrams, Inter-nat. J. Math. 8(4) (1997)537–553. Preprint arXiv:q-alg/9605028.

[Var] A. Varchenko, Special functions, KZ type equations and representation theory,CBMS 98, AMS (2003). arXiv:math.QA/0205313, May 2002.

[Va1] V. A. Vassiliev, Cohomology of knot spaces, Theory of Singularities and Its Ap-plications (ed. V. I. Arnold), Advances in Soviet Math., 1 (1990) 23–69.

[Va2] V. A. Vassiliev, Homological invariants of knots: algorithm and calculations,Preprint Inst. of Applied Math. (Moscow), No. 90 (1990), 23 pp. (in Russian).

[Va3] V. A. Vassiliev, Complements of Discriminants of Smooth Maps: Topology and

Applications, Revised ed. Providence RI: AMS, 1994. (Transl. of Math. Mono-graphs, 98).

360 Bibliography

[Va4] V. A. Vassiliev, On combinatorial formulas for cohomology of spaces of knots,Moscow Mathematical Journal, v. 1, no. 1, pp.91–123 (2001). Preprint version:http://www.pdmi.ras.ru/~arnsem/papers/), September 2000.

[Va5] V. A. Vassiliev, Combinatorial computation of combinatorial formulas for

knot invariants, Preprint, August 2001, updated September 2003, 83 pp.http://www.pdmi.ras.ru/~arnsem/papers/).

[Va6] V. A. Vassiliev, On invariants and homology of spaces of knots in arbitrary man-

ifolds, in Topics in quantum groups and finite-type invariants, (B. Feigin and V.Vassiliev, eds.) AMS Translations Ser. 2 vol. 185 (1998) 155-182, Providence.

[Vo] P. Vogel, Algebraic structures on modules of diagrams, Institut de Mathe-matiques de Jussieu, Prepublication 32, August 1995, Revised in 1997. Online athttp://www.math.jussieu.fr/~vogel/.

[Wal] W. Walter, Ordinary Differential Equations, Springer, GTM-192, 1998.

[Wee] J. Weeks, SnapPea, Computer program for investigation of hyperbolic 3-manifolds, knots and links. http://geometrygames.org/SnapPea/.

[Wil1] S. Willerton, The Kontsevich integral and algebraic structures on the space of

diagrams. Knots in Hellas ’98. Series on Knots and Everything, vol. 24, WorldScientific, 2000, 530–546. arXiv:math.GT/9909151.

[Wil2] S. Willerton, An almost integral universal Vassiliev invariant of knots.

arXiv:math.GT/0105190.

[Wit1] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys.121 (1989), 351–399.

[Wit2] E. Witten, Chern-Simons gauge theory as a string theory,arXiv:hep-th/9207094.

[Zag1] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind

eta-function, Topology 40(5) (2001) 945–960.

[Zag2] D. Zagier, The remarkable dilogarithm, J. Math. Phys. Sci. 22 (1988) 131–145.

[Zag3] D. Zagier, Values of zeta functions and their applications, First European Con-gress of Mathematics, Progr. in Math., 120, Birkhauser, Basel, II (1994) 497–512.

[Zh] D. P. Zhelobenko, Compact Lie groups and their representations, Moscow, Nauka,1970 (AMS translation, Providence, 1973).

[Zul] L. Zulli, A matrix for computing the Jones polynomial of a knot, Topology, 34

(1995) 717–729.

Notations

Z, Q, R, C — rings of integer, rational, real and complex numbers.

F — ground field (usually Q or C).

A(n), B(n)

ψn (p. 227) — n-th cabling of a knot.

τ (inverse of χ)

φg, φVg

ρg, ρVg

X — algebra of knots

I — algebra of knot invariants

symb(v) — symbol of the Vasiliev invariant v

Vn — space of Vassiliev invariants of degree ≤ n∆n

———————–

Add index of open problems

361

Index

BNG, 311

E6, 334

E7, 334

E8, 334

F4, 334

G2, 334

I′(K), 194

I(K), 193

L, 274

P (L), 49

R, 278

R,R−1, 200

R-matrix, 44

SA, 306

Si, 218, 279

SMM, 306

T -chord diagram id, 279

Z(K), 191

Ah(n), 256

An, 90

C〈〈A,B〉〉, 259

C, 38

CDn, 112

C, 106

CDn, 112

Cn, 106

Γ, 135

G, 321

K, 26

Kn, 58

L, 322

Λ, 146

Λ(L), 50

Li2, 216

MM, 305

ODn, 119

Pn, 97

Φ, 279

ΦKZ, 259

Vn, 60

W ′n, 83

Wn, 83

αn, 66

1n, 95

χ, 122

δ, 337

ε, 337

η(p,q), 267

ι, 336

max, 278

min, 278

µ, 336

∇, 60

A, 239

A′, 191

ψn, 227

slN , 333

spN , 333

soN , 333

θ(K), 49

θfr(K), 45eψn, 233

f , 264

3-graph, 134

bubble, 137

dodecahedron, 138

wheel, 138

ad-invariant

bilinear form, 334

Adjoint representation, 334

Alexander polynomial, 37

363

364 Index

Alexander–Conway polynomial, 37

Algebra, 336

horizontal chord diagrams, 256

of 3-graphs, 135

of knot invariants, 41

of knots, 27

of Vasiliev invariants, 68

universal enveloping, 336, 338

Vogel’s Λ, 146

Antipode, 345

Antisymmetry relation, 108

Associativity tangle, 217

Associator

axiomatic definition, 283

rational, 286

Baguette diagram, 325

Bar-Natan–Garoufalidis function, 311

Bialgebra, 337

connected, 345

d-filtered, 343

filtered, 343

graded, 344

i-filtered, 343

of finite type, 345

of graphs, 321

Borromean

link, 18

Borromean link, 18

Braid, 22

fundamental, 30

generators, 23

pure, 23, 257

relation, 23

relations, 23

Burau representation, 31

reduced, 31

Cabling

of chord diagrams, 229

of closed diagrams, 229

of invariants, 228

of knots, 227

of open diagrams, 229

of the Kontsevich integral, 232

of weight systems, 231

Canonical

invariant, 240

series, 240

Casimir element, 152

Casson invariant, 62

Chinese characters, 119

Chord

isolated, 83

Chord diagram, 64

coproduct, 93

of a singular knot, 65

product, 91

tangled, 198

multiplication, 279

Closed diagram, 105

connected, 115

coproduct, 115

product, 114

Coalgebra, 337

Cocommutativity, 337

Commutativity, 337

Conjecture

Melvin–Morton, 305

Tait, 20, 53

Convergent monomial, 263

Conway polynomial, 38, 243

symbol, 77, 130, 243

Coorientation, 57

Coproduct

in A, 93

in B, 120

in C, 115

Deframing



Degree, 64, 67, 105, 119, 134

Diagram

1-3-valent, 119

arrow, 298

caterpillar, 132

closed, 105

fixed, 145

Gauss, 295

Jacobi, 119

open, 119

Pont-Neuf, 329

web, 119

Dilogarithm, 216

Divergent monomial, 263

Double point, 57

resolution, 59

Drinfeld associator, 259

Duality relation, 272

Edge product, 135

Embedding, 15

Filtered basis, 342

Filtered space

reduced, 342

Four-term relation

for chord diagrams, 82

for graphs, 322

for knots, 85

for tangles, 198

Framed knot, 28

Framing, 28

Index 365

independence, 86

Framing independence, 83

Free Lie algebra L, 274

Goussarov, 67

group, 319

Graded completion, 96, 191

Graded space, 342

associated with a filtration, 342

Grading

second in B, 120

Granny knot, 18

Group algebra, 338

Group-like element, 97, 339

Hexagon relation, 284

HOMFLY polynomial, 49, 242

Homogeneous components, 345

Hopf algebra, 344

dual, 345

connected, 345



of finite type, 345



Hopf link, 18

Hump, 193

IHX relation, 108

Intersection graph, 86

conjecture, 87, 227

Intersection number, 35

Isomorphism

A ≃ C, 112

B ≃ C, 122

Isotopy, 16

Jones polynomial, 40

colored, 241

symbol, 71, 130

Kauffman bracket, 40

Kauffman polynomial, 50

Killing form, 334

Kirchhoff law, 110

generalized, 111

Knizhnik–Zamolodchikov

equation, 254

Knizhnik-Zamolodchikov

associator, 259

Knot, 15

alternating, 20

ambient equivalence, 17

ambient isotopy, 17

amphicheiral, 20, 224

figure eight, 18

framed, 28

granny, 18

invertible, 20

Morse, 191

ribbon, 28

singular, 58

square, 18

trefoil, 18

virtual, 296

Knot diagram, 18

alternating, 20

reducible, 20

Knot invariant, 33

finite type, 59

group-like, 88

primitive, 88

Kontsevich integral, 191

combinatorial, 278

convergence, 196

final, 193

for braids, 199

for tangle complement, 209

for tangles, 199

framed, 237

group-like, 215

invariance, 197

mutation, 225

of the unknot, 244

preliminary, 193

reality, 221

tangle-reduced, 206

Krammer–Bigelow representation, 31

KZ equation, 254

formal, 257

reduced, 258

Landen connection formula, 218

Lando

graph algebra, 322

Lie algebra

classical, 333

exceptional, 333

free L, 274

metrized, 335

weight systems, 151

Link, 18

Borromean, 18

Hopf, 18

trivial, 18

Whitehead, 18

Linking number, 34, 190

Mapping f , 264

Markov moves, 23

Multiple polylogarithm, 268

Multiple zeta values, 267

Multivariate ζ-function, 267

366 Index

Mutation, 225

MZV, 262, 267

One-term relation, 83

Open diagram, 119

Operation

Si on T -chord diagrams, 218, 279

∆i on T -chord diagrams, 279

εi on T -chord diagrams, 280

Order, 64, 105, 119, 134

in a d-filtered space, 342

in an i-filtered space, 342

Orientation

detecting, 117, 129

Parenthesized presentation of a knot, 275

Pentagon relation, 283

Polylogarithm, 268

Primitive element, 339

Primitive space

dimensions, 116

filtration, 116

in A, 97

in C, 115

Product

in A, 91

in B, 120

in C, 114

in Γ, 135, 141

of tangles, 22

Quantum group, 43

Quantum invariant, 241

framed, 45

unframed, 49

Quillen, 347

Reidemeister moves, 19

framed, 28

Relation

AS, 108

four-term, 82

IHX, 108

one-term, 83

STU, 106

Resolution

complete, 59

Ribbon graph, 134

Ribbon knot, 28

Roger five-term relation, 218

Rotation, 106

Scheme, 326

Self-linking number, 37

Semigroup-like element, 339

shuffle, 215

Singularities, 56

Skein relation

Conway’s, 38

Square knot, 18

String link, 22

STU, 106

Symbol, 66

T-diagram, 168

Tangle, 21

associativity, 217

elementary, 24

product, 22

simple, 24

tensor product, 22

Theorem

Alexander, 23

Birman–Lin, 74

Goussarov–Habiro, 313

Le–Murakami–Kassel, 241

Markov, 23

Milnor–Moore, 346

Quillen, 347

Reidemeister, 19

framed, 28

Turaev, 24

Vassiliev–Kontsevich, 84

Trefoil, 18

Turaev moves, 24

Unframed chord diagrams, 90

Univariate zeta function, 274

Universal enveloping algebra, 336

Universal Vassiliev invariant, 211

Unknot, 38

Kontsevich integral, 244

Vasiliev invariants

algebra of, 68

Vassiliev

extension, 58

invariant, 59

canonical, 214, 240

framed, 86

of order 2, 62

skein relation, 58

Vassiliev invariant

group-like, 88

primitive, 88

universal, 211

Vassiliev’s

extension

as differentiation, 60

Vector space




of unframed chord diagrams, 90

Index 367

Vertex

external, 108internal, 108

Vertex product, 141Vogel

algebra Λ, 146

Weightof a MZV, 273

Weight system, 81

glN , 157, 167sl2, 154, 165slN , 159soN , 161, 169

sp2N , 162Lie algebra, 151multiplicative, 97of the Conway coefficients, 184of the Jones coefficients, 157, 182unframed, 83

Wheel, 116Wheeling, 244Wheels formula, 244Whitehead link, 18Whitney trick, 32Wilson loop, 106Witten, 9Writhe

local, 19of a framed knot, 239of a knot diagram, 19total, 19

Yang–Baxter equation, 44

cdbook · §4.5. hopf algebra of chord diagrams 90 §4.6. hopf algebra of weight systems 95 §4.7....

Documents