zsuzsanna dancso- on a universal finite type invariant of knotted trivalent graphs

8/3/2019 Zsuzsanna Dancso- On a universal finite type invariant of knotted trivalent graphs

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On a universal finite type invariant of knotted trivalent

graphs

by

Zsuzsanna Dancso

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Mathematics

University of Toronto

Copyright c 2011 by Zsuzsanna Dancso


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Abstract

On a universal finite type invariant of knotted trivalent graphs

Zsuzsanna Dancso

Doctor of Philosophy

Graduate Department of Mathematics

University of Toronto

2011

Knot theory is not generally considered an algebraic subject, due to the fact that knots

dont have much algebraic structure: there are a few operations defined on them (such

as connected sum and cabling), but these dont nearly make the space of knots finitely

generated. In this thesis, following an idea of Dror Bar-Natans, we develop an algebraic

setting for knot theory by considering the larger, richer space of knotted trivalent graphs

(KTGs), which includes knots and links. KTGs along with standard operations defined

on them form a finitely generated algebraic structure, in which many topological knot

properties are definable using simple formulas. Thus, a homomorphic invariant of KTGs

provides an algebraic way to study knots.

We present a construction for such an invariant. The starting point is extending

the Kontsevich integral of knots to KTGs. This was first done in a series of papers by

Le, Murakami, Murakami and Ohtsuki in the late 90s using the theory of associators.

We present an elementary construction building on Kontsevichs original definition, and

discuss the homomorphicity properties of the resulting invariant, which turns out to be

homomorphic with respect to almost all of the KTG operations except for one, called

edge unzip. Unfortunately, edge unzip is crucial for finite generation, and we prove

that in fact no universal finite type invariant of KTGs can intertwine all the standard

operations at once. To fix this, we present an alternative construction of the space of

KTGs on which a homomorphic universal finite type invariant exists. This space retains

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all the good properties of the original KTGs: it is finitely generated, includes knots, and

is closely related to Drinfeld associators.

The thesis is based on two articles, one published [Da] and one preprint [BD1], the

second one joint with Dror Bar-Natan.

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Acknowledgements

I would like to express my deepest gratitude to my advisor, Dror Bar-Natan, from

whom I learned how to do and enjoy mathematics, who spent large amounts of his time

working with me, and seemed to have believed in me throughout. Like most graduate

students probably do, I have hit a few walls in the past five years: got stuck on problems,

got discouraged, have lost focus or enthusiasm, and each time Dror tried idea after idea to

get me mathematically, psychologically, technologically and generally un-stuck. He spent

countless hours discussing mathematics with me, encouraged me to attack difficulties I

had given up on, suggested new directions when projects hit dead ends, told me that he

thought I could make it in academia when I did not think I could, waited paitently when I

was preoccupied with life outside mathematics, introduced me to other mathematicians,

pushed me to give talks and taught me a lot about how to do it better, gave me many

computer tutorials, funded my travels, and has overall been a spectacular advisor. As

much as it still seems like a bit of a miracle, he may have succeeded, and I will never be

able to thank him enough. I believe that choosing to work with Dror was one of the best

choices I have ever made. Dror- thank you for everything.

Without doubt I would never have made it here without the love and support of my

husband Balazs, also a mathematician, and also one of my best choices ever made. I

met Balazs shortly after starting my undergraduate studies in math, and he has been

a limitless source of inspiration, encouragement and warmth ever since, even at times

when I did not deserve it. Among many other gifts, he has given me the best piece of

graduate school advice I have ever recieved: in my first year in Toronto I was complaining

to him about my doubts regarding academia and what an academic career entails, and

whether I am good enough to even try. He told me that was nonsense, that graduate

school is a fabulous time of ones life, a rare combination of relative freedom finanncially

and in terms of responsibilities, an opportunity to make good friends and spend time and

share experiences with them, a time to do interesting research and learn new things and

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meanwhile have fun. I took this advice to heart, and this made for a (possibly unusually)

smooth and happy graduate school experience, during which Balazs and I made many

happy memories. Bali- thank you for being you and loving me.

I am lucky and grateful to have the most wonderful family one could wish for: my

parents, who have always encouraged me to strive for achievement; my grandmother,

who will undoubtedly be the biggest fan of this thesis, albeit not the most expert one;

and my brother and sister, who can always make me laugh. Anyu, Apu, Nyonyo, Marcsi,

Tomi- thank you for the childhood where I am always happy to return.

Last but not least, I would like to thank all of my fantastic friends, old and new, for

always being there. Thank you for all you have done with me, for me, or because of me-

you know who you are.

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Contents

1 Introduction 1

2 Preliminaries 5

2.1 Finite type invariants- the classical view . . . . . . . . . . . . . . . . . . 5

2.2 Finite type invariants- the algebraic view . . . . . . . . . . . . . . . . . . 10

2.2.1 Algebraic structures and expansions . . . . . . . . . . . . . . . . . 10

2.2.2 Knotted trivalent graphs . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 The definition of the Kontsevich Integral . . . . . . . . . . . . . . . . . . 22

2.3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 The Kontsevich integral of knotted trivalent graphs 34

3.1 The naive extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.1 The good properties . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.2 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Eliminating the divergence . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 The re-normalized integral Z1 . . . . . . . . . . . . . . . . . . . . 37

3.2.2 The good properties . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Corrections - constructing a KTG invariant . . . . . . . . . . . . . . . . . 48

3.3.1 Missing moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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3.3.2 Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3.3 Translating to equations . . . . . . . . . . . . . . . . . . . . . . . 50

3.3.4 The resulting invariants . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Non-homomorphicity: unzip . . . . . . . . . . . . . . . . . . . . . . . . . 61

4 Fixing unzip: dotted KTGs 65

4.1 The space of dotted KTGs . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.2 The associated graded space and homomorphic expansion . . . . . . . . . 68

4.3 An equivalent construction . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Applications 75

5.1 The relationship with Drinfeld associators . . . . . . . . . . . . . . . . . 75

5.2 A note on the Kirby band-slide move and the LMO invariant . . . . . . . 89

Bibliography 90

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Chapter 1

Introduction

The goal of this thesis is to construct an algebraic setting for studying knot theory. In

itself, knot theory is not very algebraic: the space of knots has few operations defined

on it (most notably connected sum and cabling), and these are not nearly enough to

make knots finitely generated. However, we can embed knots in a larger, richer space

of knotted trivalent graphs (KTGs), which, along with standard operations defined on

them, form a finitely generated algebraic structure. Our objective is to construct a

homomorphic expansion (or universal finite type invariant) of KTGs: an invariant with

a strong universality property which is very well-behaved under all KTG operations in

the appropriate sense. This idea is due to Dror Bar-Natan, and was raised in [BN2].

One reason this approach is of interest is that several knot properties (such as knot

genus, unknotting number and ribbon property, for example) are definable by short for-

mulas involving knotted trivalent graphs and the aforementioned operations. Therefore,

such an operation-respecting invariant yields algebraic necessary conditions for these

properties, i.e. equations in the target space of the invariant. The extension of Z pre-

sented in Chapters 3 and 4 is the first example for such an invariant. Unfortunately, the

target space of Z is too complicated for it to be useful in a computational sense. How-

ever, we hope that by finding sufficient quotients of the target space more computable

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Chapter 1. Introduction 2

invariants can be born.

The first step of our construction is to extend the Kontsevich integral Z of knots to

knotted trivalent graphs. The extension is a universal finite type invariant of knotted

trivalent graphs, which is almost homomorphic: intertwines almost all standard opera-

tions defined on the space of graphs with their counterparts induced on the target space.

These are changing the orientation of an edge; deleting an edge; unzipping an edge (an

analogue of cabling of knots); and connected sum.

The Kontsevich invariant Z was first extended to knotted trivalent graphs by Mu-

rakami and Ohtsuki in [MO], and later by Cheptea and Le in [CL]. Both papers extend

the combinatorial definition of Z, using q-tangles (a.k.a. parenthesized tangles) and

building on a significant body of knowledge about Drinfelds associators (even associa-

tors in [CL]) to prove that the extension is a well-defined invariant.

In Chapter3 we try to extend Z naively, replacing knots by knotted trivalent graphs

in the original definition. Unfortunately, the result is neither convergent nor an isotopy

invariant. Thus, one needs to apply re-normalizations to make it convergent, and correc-

tions to make it invariant. Using q-tangles, [MO] and [CL] do not have to deal with the

convergence issue, while similar invariance issues arise in both approaches.

The main purpose of this approach to extending the Kontsevich integral is to eliminate

the black box quality of the previous constructions, which results partly from the depth

of the ingredients that go into them, and partly from the fact that the proofs are spread

over several papers ([MO, LMMO, LM] or [CL, MO]).

Our construction differs from previous ones in that we build the corrected extension

step by step on the naive one. After re-normalizations to make the extension convergent

(a baby version of re-normalization used in quantum field theory), non-invariance errors

arise. We fix some of these by introducing counter terms (corrections) that are precisely

the inverses of the errors, and we show that thanks to some syzygies, i.e. dependencies

between the errors, all the other errors get corrected automatically. This is where the


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bulk of the difficulty lies. The proofs involve mainly elementary Kontsevich integral

methods and combinatorial considerations.

The problem with the extended invariant, which we have alluded to before, is that

it is only almost a homomorphism: it fails to commute with the unzip operation, which

plays a crucial role in the finite generation of KTGs. Although its behavior with respect

to unzip is well-understood, it is not homomorphic, and we show that this is not a

shortcoming of this sepcific construction: any expansion (universal finite type invariant)

of KTGs will display a similar anomaly, i.e., can not commute with all four operations

at the same time.

The goal of Chapter 4 is to fix this problem by proposing a different definition of

KTGs, which we will call dotted knotted trivalent graphs, or dKTGs. These form a

finitely generated algebraic structure on which a truly homomorphic expansion exists. We

present two (equivalent) constructions of this space. In one we replace the unzip, delete

and connected sum operations by a more general set of operations called tree connected

sums. In the other, we restrict the set of edges which we allow to be unzipped. We

show that the invariant of Chapter 3 can easily be modified to produce a homomorphic

expansion of dKTGs.

In particular, this provides an algebraic description of the Kontsevich integral (of

knots and graphs), due to the fact that dKTGs are finitely generated, i.e. theres a finite

set (of size 4) of dKTGs such that any dKTG can be obtained from these using the above

operations. This is described for KTGs in more detail in [Th], and is easily modified to

dKTGs in Chapter 4. Sine our extension of the Kontsevich integral commutes with all

operations, it is enough to compute it for the graphs in the generating set. As knots are

special cases of knotted trivalent graphs, this also yields an algebraic description of the

Kontsevich integral of knots.

Finally, in Chapter 5 we describe two applications. The first is the connection to

Drinfeld associators, in particular, the construction of an associator as the value of the


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invariant on a tetrahedron graph. The second is a simple (free of associators and local

considerations) proof of the (recently falsely disputed) theorem that the LMO invariant

is well behaved with respect to the Kirby II (band-slide) move.


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Chapter 2

Preliminaries

2.1 Finite type invariants- the classical view

The main reference for this section is Chmutov and Duzhins expository paper on the

Kontsevich integral, [CD].

The theory of finite type (or Vassiliev) invariants grew out of the idea of V. Vassiliev

to extend knot invariants to the class of singular knots. By a singular knot we mean

a knot with a finite number of simple (transverse) double points. The extension of a

C-valued knot invariant f follows the rule

f( ) = f( ) f( ).

A finite type (Vassiliev) invariant is a knot invariant whose extension vanishes on all

knots with more than n double points, for some n N. The smallest such n is called the

order, or type of the invariant.

The set of all Vassiliev invariants forms a vector space V, which is filtered by the

subspaces Vn, the Vassiliev invariants of order at most n:

V0 V1 V2 ... Vn ... V

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Chapter 2. Preliminaries 6

This filtration allows us to study the simpler associated graded space:

grV = V0 V1/V0 V2/V1 ... Vn/Vn1 ...

The components Vn/Vn1 are best understood in terms of chord diagrams.

A chord diagram of order n is an oriented circle with a set of n chords

all of whose endpoints are distinct points of the circle (see the figure on the

right). The actual shape of the chords and the exact position of endpoints

are irrelevant, we are only interested in the pairing they define on the 2n cyclically ordered

points, up to orientation preserving diffeomorphism.

The chord diagram of a singular knot S1 S3 is the oriented circle S1 with the

pre-images of each double point connected by a chord.

Let Cn be the vector space spanned by all chord diagrams of order n, and let Fn be

the vector space of all C-valued linear functions on Cn. A Vassiliev invariant f Vn

determines a function [f] Fn defined by [f](C) = f(K), where K is some singular knot

whose chord diagram is C.

The fact that [f] is well defined (does not depend on the choice of K) can be seen

as follows: IfK1 and K2 are two singular knots with the same chord diagram, then they

can be projected on the plane in such a way that their knot diagrams coincide except

possibly at a finite number of crossings, where K1 may have an over-crossing and K2 an

under-crossing or vice versa. But since f is an invariant of order n and K has n double

points, a crossing flip does not change the value of f (since the difference would equal to

the value of f on an (n + 1)-singular knot, i.e. 0).

The kernel of the map Vn Fn is, by definition, Vn1. Thus, what we have defined is

an inclusion in : Vn/Vn1 Fn. The image of this inclusion (i.e. the set of linear maps

on chord diagrams that come from Vassiliev invariants) is described by two relations,

the four-term, or 4T, and the framing independence, or F I relation, also known as the

one-term relation.


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The dotted arcs in the pictures that follow mean that there might be further chords

attached to the circle, the positions of which are fixed throughout the relation.

The four-term relation asserts that

f( ) f( ) + f( ) f( ) = 0,

for an arbitrary fixed position of (n 2) chords (not drawn here) and the two additional

chords as shown.

This follows from the following fact of singular knots:001101 0 01 10 00 01 11 1 0011010 01 10 01 1f( ) + f( ) + f( ) + f( ) = 0,which is easy to show using the isotopy below:

= = .

The framing independence relation arises from the Reidemeister 1 move of knot dia-

grams:

f( ) = 0,

To explain the name of this relation and because we will need this later, let us take

a short detour to talk about framed knots. A framing on a curve is a smooth choice of a

normal vector at each point of the curve, up to isotopy. This is equivalent to thickening

the curve into a band, where the band is always orthogonal to the chosen normal vector.

Note that due to the smooth choice of normal vectors, the band has an even number

of twists on it, and thus forms an oriented surface. Observe that framed knots are a

Z-extension of knots, as the number (with sign) of double twists determines the framing

up to isotopy.


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A knot projection (knot diagram) defines a framing (blackboard fram-

ing), where we always choose the normal vector that is normal to (and

points up from) the plane that we project to. Every framed knot (up to isotopy) can

be represented as knot projection with blackboard framing, as adding a double twist

amouns to adding a kink to the diagram, as shown in the figure on the right.

Of the Reidemeister moves, R2 and R3 do not change the blackboard framing of a

knot diagram. R1, however, adds or eliminates a kink, and hence changes the framing.

However, by dropping R1, we get something slightly larger than framed knots: R2 and

R3 also leave the rotation number of the knot projection fixed, while R1 does not. In

fact, there are four types of R1 moves, depending on whether the kink is to the left or to

the right of the strand and whether the crossing is positive or negative, as shown below.

R+ R

or or or

L+ L

Left/right affects the rotation number and +/ affets the framing. Indeed, one can

verify easily that a succession ofL+ and R or L and R+ can be trivialized using only

R2 and R3. Hence, knot projections modulo R2 and R3 form a Z2- extension of knots,

where framing accounts for one factor ofZ and rotation number accounts for the other.

R1To get rid of rotation number and obtain a characterization of framed

knots, one needs R2, R3 and an additional relation which we will call R1,

which lets us eliminate an L+ followed by an L, as shown on the right.

(This is eqvivalent, via R2 and R3, to eliminating an R+ followed by R.) The same

can be said for links, with the difference that both framing and rotation number adds a

factor ofZ for each link component.

The framing independence relation we work with here arises from R1, meaning that

the knot invariants we work with are independent of any framing on the knot. This will

change later as we turn to (framed) knotted trivalent graphs.


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We define the algebra A, as a direct sum of the vector spaces An generated by chord

diagrams of order n considered modulo the FI and 4T relations. By an abuse of notation,

from now on we consider 4T and F I to be relations in Cn, i.e. define the appropriate

(sums of) chord diagrams to be zero. The multiplication on A is defined by the connected

sum of chord diagrams, which is well defined thanks to the 4T relations (for details, see

for example [BN1]).

C-valued linear functions on An are called weight systems of order n. The above

construction shows that every Vassiliev invariant defines a weight system of the same

order.

The famous theorem of Kontsevich, also known as the Fundamental Theorem of Finite

Type Invariants, asserts that every weight system arises as the weight system of a finite

type invariant. The proof relies on the construction of a universal finite type invariant,

by which we mean a knot invariant which takes its values in the graded completion of A

with the property that when evaluated on a singular knot, the lowest order term of the

result is the corresponding chord diagram. Note that a universal finite type invariant isnot a finite type invariant, but rather encompasses all the information contained in all

finite type invariants. The (essentially unique) universal finite type invariant of knots

is the Kontsevich Integral Z, the main ingredient of this thesis. To prove Kontsevichs

Theorem, given any weight system, one gets the appropriate finite type invariant (up to

invariants of lower type) by pre-composing it with Z.

Our overall goal is to develop an algebraic context for the subject by embedding

knots in the finitely generated space of (dotted) knotted trivalent graphs and extending

the Kontsevich integral to obtain a universal finite invariant of this space, which also

respects the algebraic structure (is homomorphic). Let us begin by introducing a

general notion of algebraic structures and expansions, and re-introducing finite type

invariants from this point of view.


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2.2 Finite type invariants- the algebraic view

The goal of this section is to introduce the theory of finite type invariants by putting it in

the general algebraic context of expansions. We first define general algebraic structures,

projectivizations and expansions, followed by a short introduction to finite type

invariants of knotted trivalent graphs (including the special case of knots and links) as

an example: knotted trivalent graphs form a finitely presented algebraic structure which

includes knots and links, the projectivization of this structure is the corresponding space

of chord diagrams (a graded space), and the word expansion translates to universal

finite type invariant. The presentation follows [BD1] and [BN5].

2.2.1 Algebraic structures and expansions

An algebraic structure O is some collection (O) of sets of objects of different kinds,

where the subscript denotes the kind of the objects in O, along with some collection

ofoperations , where each is an arbitrary map with domain some product O1

Ok of sets of objects, and range a single set O0 (so operations may be unary or binary

or multi-nary, but they always return a value of some fixed kind). We also allow some

named constants within some Os (or equivalently, allow some 0-nary operations). The

operations may or may not be subject to axioms an axiom is an identity asserting

that some composition of operations is equal to some other composition of operations.

Any algebraic structure O has a projectivization. First extend O to allow formal

linear combinations of objects of the same kind (over some field, here we will work

over Q), extending the operations in a linear or multi-linear manner. Then let I, the

augmentation ideal, be the sub-structure made out of all such combinations in which the

sum of coefficients is 0. Let Im be the set of all outputs of algebraic expressions (that

is, arbitrary compositions of the operations in O) that have at least m inputs in I (and


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possibly, further inputs in O), and finally, set

proj O :=m0

Im/Im+1.

Clearly, with the operations inherited from O, the projectivization proj O is again alge-

braic structure with the same multi-graph of spaces and operations, but with new objects

and with new operations that may or may not satisfy the axioms satisfied by the opera-

tions ofO. The main new feature in proj O is that it is a graded structure, the degree

m piece being Im/Im+1.

Given an algebraic structure O let fil O denote the filtered structure of linear combina-

tions of objects in O (respecting kinds), filtered by the powers (Im) of the augmentation

ideal I. Recall also that any graded space G =

m Gm is automatically filtered, bynm Gn

m=0

.

An expansion Z for O is a map Z : O proj O that preserves the kinds of objects

and whose linear extension (also called Z) to fil O respects the filtration of both sides,

and for which (gr Z) : (gr O = proj O) (gr proj O = proj O) is the identity map of

proj O.

In practical terms, this is equivalent to saying that Z is a map O proj O whose

restriction to Im vanishes in degrees less than m (in proj O) and whose degree m piece

is the projection Im Im/Im+1.

A homomorphic expansion is an expansion which also commutes with all the alge-

braic operations defined on the algebraic structure O.

2.2.2 Knotted trivalent graphs

A trivalent graph is a graph which has three edges meeting at each vertex. We require

that all edges be oriented and that vertices be equipped with a cyclic orientation, i.e. a

cyclic ordering of the three edges meeting at the vertex. We allow multiple edges; loops

(i.e., edges that begin and end at the same vertex); and circles (i.e. edges without a


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vertex).

2

3

2

3

1 1

Given a trivalent graph , its thickening is obtained

from it by thickening verices as shown on the right, and

gluing the resulting thick Ys in an orientation preserving

manner, where the cyclic orientation at the vertex defines the orientation of the surface.

Hence, is a two-dimensional oriented1 surface with boundary.

A Knotted Trivalent Graph (KTG) is an isotopy class of embeddings of

a thickened trivalent graph in R3, as shown. This is equivalent to saying

that the edges of the graph are framed and the framings (normal vectors)

agree at vertices: the embedding of the thickened graph has a dark side and a white

side due to being oriented, and by definition, the framing is the normal vector to the

surface along the middle of the ribbon on the dark side, so at vertices the three normal

vectors agree. Conversely, such a framed embedding obviously determines an embedding

of the thickening, up to isotopy. As a special case, framed knots and links are knotted

trivalent graphs. The skeleton of a KTG is the combinatorial object (trivalent graph

) behind it.

Isotopy classes of KTGs also have a characterisation in terms of graph projections

(graph diagrams) and Reidemeister moves. Graph diagrams are projections onto a plane

with only transverse double points preserving the over- and under-strand information at

the crossings. Unframed KTGs correspond to graph diagrams modulo the Reidemeister

moves R1 R2, R3 and R4 (see for example [MO]). R1, R2 and R3 are the same as in

the knot/link case. R4 involves moving a strand in front of or behind a vertex:

R4b :R4a :

Since we are considering framed KTGs, we need a charcterisation of these, which is

1There is a slightly different non-oriented version which we do not describe in detail here.


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similar to that of framed knots:

Proposition 2.2.1. KTGs (according to our definition, framed) are in one-to-one corre-

spondence with graph diagrams modulo Reidemeister moves R1, R2, R3 and R4, where

R1 eliminates adjacent positive and negative left kinks, as in the case of framed knots.

Proof sketch. We understand a graph diagram to come with the blackboard framing.

First we show that any isotopy class of KTGs can be represented by a blackboard framed

graph diagram. Choose a plane to project to, and deform small neighborhoods of the

vertices to flat Ys parallel to the plane. Now each edge has a (not necessarily even)

integer number of twists on it. The goal is to have even numbers on each edge, since a

double twist can be represented by a kink in the blackboard framed projection.root

leaves

pushing twists off the tree

To achieve this, fix a spanning tree of the graph and

iron it: align it with the blackboard framing by moving

all twists off the tree, starting from the root and progressing

towards the leaves, using the isotopy pictured on the right.

The goal is for the tree to be flat ribbon and for all the twists to lie on the curves

connecting the leaves of the ironed spanning tree. Since the thickened graph is an oriented

surface, each of these curves contain an even number of twists. These can now be

represented on the projection as kinks.root

leavesmoving kinks off the tree

R2, R3, R4

Next, we need to prove that isotopies of KTGs can

be realised by Reidemeister moves, the other direction

being obvious. Suppose two KTGs, represented by

blackboard framed graph diagrams, are isotopic. Fix

a spanning tree of the common skeleton. We leave it

to the reader to convince him- or herself that one can use R2, R3 and R4 to bring the

spanning trees into identical positions, move shrink kinks to a small size and move them

off the trees. The moving of kinks, in particular, can be done by the move shown on the

right, which is a combination of R2, R3 and R4 moves.


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Now one can again use R2, R3 and R4 to bring the arcs connecting the ends of the

spanning trees into the same position, so the only difference is the number of kinks on

the arcs. Kinks move freely on the strands and commute with each other using R2, R3

and R4, and if the graphs are isotopic, then the framing, determined by the signed count

of kinks on each arc, is the same. So one can use R1 to cancel all the kinks that can be

cancelled, and the remaining essential ones can be brought into the same position.

As an algebraic structure, KTGs have a different kind of objects for each skeleton.

The sets of objects are the sets of knottings K() for each skeleton graph . There are

four kinds of operations defined on KTGs:

Given a trivalent graph , or a knotting K(), and an edge e of , we can switch

the orientation of e. We denote the resulting graph by Se(). In other words, we have

defined unary operations Se : K() K(Se()).

We can also delete the edge e, which means the two vertices at the ends of e also

cease to exist to preserve the trivalence. To do this, it is required that the orientations

of the two edges connecting to e at either end match. This operation is denoted by

de : K() K(de()).Unzipping the edge e (denoted by ue : K() K(ue()), see figure below) means

replacing it by two edges that are very close to each other. The two vertices at the

ends ofe will disappear. This can be imagined as cutting the band ofe in half lengthwise.

In the case of a trivalent graph , we consider its thickening and similarly cut the edge

e in half lengthwise. Again, the orientations have to match, i.e. the edges at the vertex

where e begins have to both be incoming, while the edges at the vertex where e ends

must both be outgoing.

ue()e

Given two graphs with selected edges (, e) and (, f), the connected sum of these

graphs along the two chosen edges, denoted #e,f, is obtained by joining e and f by


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a new edge. For this to be well-defined, we also need to specify the direction of the new

edge, the cyclic orientations at each new vertex, and in the case of KTGs, the framing on

the new edge. To compress notation, let us declare that the new edge be oriented from

towards , have no twists, and, using the blackboard framing, be attached to the right

side of e and f, as shown:

e f

#e,f

e f

As a short detour, we show how some topological knot properties are definable by

KTG formulas.

Theorem 2.2.2. 1. {Knots bounding a surface of genus k}= { : K( ...1 2 k

)},

where , the boundary operator, is a certain fixed composition of KTG operations.

2. {Knots of unknotting number k}= {xn() : K( ...k21

), dk() = O}, where

x, the crossing change operation is a given combination of KTG operations, dk

refers to deleting the k middle edges, and O denotes the unknot.

3. {Ribbon knots}={uk1() : K(1 2 3 k

... ), dk1() = OO...O for some

k}, where uk1 denotes unzipping the (k 1) connecting edges of the k-dumbbell

graph, while dk1 refers to deleting these same edges, and OO...O denotes a trivial

link of k components.

ribbonnot ribbon

Before proving the theorem, let us review the definition of ribbon

knots. A knot is ribbon, if it bounds a singular disk in R3 such that

all the singualrities are transverse of ribbon type, as shown in the

figure on the right. This is not the most standard definition (which is phrased in terms


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of the preimages of the singularities, which form a 1-manifold in D2), but is eqvivalent

to it.

Proof (sketch).

1. Imagining the camel graph ...1 2 k

as a band graph, i.e. an oriented sur-

face, the boundary operator turns it into its boundary: ... .

The reader can check that the genus of the k-humped camel surface is k, which

implies the statement. All that is left to show is that is a composition of KTG

operations. Morally, is unzipping all edges, but of this cant really be done:

there arent enough vertices for it. The solution is to plant a triangle at each

vertex of the camel by taking connected sum with a tetrahedron, followed by two

unzips, as shown:

u2#

After this trick, it is possible to unzip all the old edges to get exactly the boundary

we want.

2. The crossing change operation x corresponding to a given edge acts the following

way:x

. We leave it as an exercise for the reader to verify that this

can be written as a composition of KTG operations and provides a description of

the unknotting number.

3. The proof has two directions. We first show that any uk() for as stated is ribbon.

Deleting the connecting edges of the dumbbell graph produces a trivial link of k

components, therefore each of the k circles bound disjoint embedded disks. Adding

back the connecting edges, these now pass through the interiors of these disjoint

disks transversely (after possibly a small perturbation). Unzipping these edges will


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unite all the disks into one disk, with ribbon type singularities where the edges

passed through the original disks.

For the other direction, cut up the disk at each ribbon singularity:

If there are (k 1) ribbon singularities, we obtain k disjoint embedded disks.

Connecting the two sides of each cut with edges gives the dumbbell graph we are

looking for.

As an algebraic structure, KT G is finitely generated2

(see [Th]), by two elements, the trivially embedded tetrahe-

dron and the twisted tetrahedron, shown on the right (note

that these only differ in framing).

As described in the general context, we allow formal Q-linear combinations of KTGs

and extend the operations linearly. The augmentation ideal I is generated by differences

of knotted trivalent graphs of the same skeleton. KT G is then filtered by powers of

I, and the projectivization A := proj KT G also has a different kind of object for each

skeleton , denoted A().

The classical way to filter the space of KTGs, which leads to the theory of finite

type invariants, is by resolutions of singularities, as described above in the case of knots

and links. An n-singular KTG is a trivalent graph immersed in R3 with n transverse

double points. A resolution of such a singular KTG is obtained by replacing each double

point by the difference of an over-crossing and an under-crossing, which produces a linear

combination of 2k KTGs. Resolutions of n-singular KTGs generate the n-th piece of the

filtration.

2In the appropriate sense it is also finitely presented, however we do not pursue this point here.


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Theorem 2.2.3. [BD1] The filtration by powers of the augmentation ideal I coincides

with the classical finite type filtration.

Proof. Let us denote the n-th piece of the classical finite type filtration by Fn, and the

augmentation ideal by I. First we prove that I = F1.

I is linearly generated by differences, i.e., I = 1 2, where 1 and 2 are KTGs

of the same skeleton. F1 is linearly generated by resolutions of 1-singular KTGs, i.e.

F1 = , where and differ in one crossing change. Thus, it is obvious that

F1 I. The other direction, I F1 is true due to the fact that one can get to any

knotting of a given trivalent graph (skeleton) from any other through a series of crossing

changes.

To prove that In Fn, we use that I = F1. (F1)n is generated by formulas

containing n 1-singular KTGs, possibly some further non-singular KTGs, joined by con-

nected sums (the only binary operation), and possibly with some other operations (un-

zips, deletes, orientation switches) applied. The connected sum of a k-singular and an

l-singular KTG is a (k + l)-singular KTG. It remains to check that orientation switch,

delete and unzip do not decrease the number of double points. Switching the orientation

of an edge with a double point only introduces a negative sign. Unzipping an edge with a

double point on it produces a sum of two graphs with the same number of double points.

Deleting an edge with a double point on it produces zero. Thus, an element in (F1)n is

n-singular, therefore contained in Fn.

The last step is to show that Fn In, i.e., that one can write any

n-singular KTG as n 1-singular, and possibly some further non-singular

KTGs with a series of operations applied to them. The proof is in the

same vein as proving that KTGs are finitely generated [Th], as illustrated here on the

example of a 2-singular knotted theta-graph, shown on the right. In the figures, a triva-

lent vertex denotes a vertex, while a 4-valent one is a double point. As shown in the

figure below, we start by taking a singular twisted tetrahedron for each double point,


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slide u3#4

u and slide d

a (non-singular) twisted tetrahedron for each crossing, and a standard tetrahedron for

each vertex, as shown in the figure below. We then apply a vertex connected sum (the

composition of a connected sum and two unzips, as defined at the beginning of the proof

of Theorem 5.1.1) along any tree connecting the tetrahedra, followed by sliding and un-

zipping edges, as shown below. The result is the desired KTG with an extra loop around

it. Deleting the superfluous loop concludes the proof.

As in the classical theory of finite type invariants, A() is best un-

derstood in terms of chord diagrams. A chord diagram of order n on a

skeleton graph is a combinatorial object consisting of a pairing of 2n

points on the edges of , up to orientation preserving homeomorphisms of the edges.

Such a structure is illustrated by drawing n chords between the paired points, as seen

in the figure on the right. From the finite type point of view, a chord represents the

difference of an over-crossing and an under-crossing (i.e. a double point).

Chord diagrams are factored out by two classes of relations, the 4T relations:

+ = 0,

and the Vertex Invariance relations (V I), (a.k.a. branching relation in [MO]):

+ (1) + (1)(1) = 0.


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In both pictures, there may be other chords in the parts of the graph not shown, but

they have to be the same throughout. In V I, the sign (1) is 1 if the orientation of

the edge the chord is ending on is outgoing, and +1 if it is incoming.

The 4T relation is proven the same way as we have seen in the case of knots, and the

V I relation arises from a similar lassoe isotopy around a vertex.

Although it is easy to see that these relations are present, showing that there are

no more is difficult, and is best achieved by constructing an expansion (in finite type

language, a universal finite type invariant) QKT G A, first done in [MO] by extending

the Kontsevich integral Z of knots, and later in [Da], which constitutes most of Chapter

3 of this thesis. The resulting invariant will be denoted by Z2 (as it is built through a

two-step construction).

The finite type theory of knots and links is included in the above as a special case. On

knots, there is no rich enough algebraic structure for the finite type filtration to coincide

with powers of the augmentation ideal with respect to some operations. However, knots

and links form a subset of KTGs, and the restriction of In to that subset reproduces the

usual theory of finite type invariants of knots and links, and Z2 restricts to the Kontsevich

integral.

Now we turn to the question of whether Z2 is homomorphic with respect to the

algebraic structure ofKT G. To study this we first have to know the operations induced

on A by Se, de, ue and #e,f.

Given a graph and an edge e, the induced orientation switch operation is a linear

map Se : A() A(se()) which multiplies a chord diagram D by (1)k where k is the

number of chords in D ending on e. This is due to the fact that switching the orientation

of an edge turns an under-crossing into an over-crossing and vice versa. Note that this

generalizes the antipode map on Jacobi diagrams, which corresponds to the orientation

reversal of knots (see [Oh], p.136).

The induced edge delete is a linear map de : A() A(de()), defined as follows:


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when the edge e is deleted, all diagrams with a chord ending on e are mapped to zero

(an over- and an under-crossing become the same when one of the participating edges

is deleted), while those with no chords ending on e are left unchanged, except the edge

e is removed. Edge delete is the generalization of the co-unit map of [Oh] (p.136), and

[BN1].

The induced unzip is a linear map ue : A() A(ue()). When e is unzipped,

each chord that ends on it is replaced by a sum of two chords, one ending on each new

edge (i.e., if k chords end on e, then ue sends this chord diagram to a sum of 2k chord

diagrams).

There is an operation on A(O) corresponding to the cabling of knots: references

include [BN1] (splitting map) and [Oh] (co-multiplication). The graph unzip operation

is the graph analogy of cabling, so the corresponding map is analogous as well.

For graphs and , with edges e and e, the induced connected sum #e,e : A()

A() A(#e,e) acts in the obvious way, by performing the connected sum operation

on the skeletons and not changing the chords in any way. This is well defined due to

the 4T and V I relations. (What needs to be proven is that we can move a chord ending

over the attaching point of the new edge; this is done in the same spirit as the proof of

Lemma 3.1 in [BN1], using hooks; see also [MO], figure 4.)

u(1/2

)

1/2 1/2

Z2() Z2(u())

As it turns out (see [MO, Da]), Z2 is almost homomor-

phic: it intertwines the orientation switch, edge delete, and

connected sum operations. However, Z2 does not commute

with edge unzip. The behavior with respect to unzip is well-

understood (showed in [Da] using a result of [MO]), and is

described by the formula shown in the figure on the right.

Here, denotes the Kontsevich integral of the un-knot. A formula for was conjectured

in [BGRT1] and proven in [BLT]. The new chord combinations appearing on the right

commute with all the old chord endings by 4T. A different way to phrase this formula


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is that Z2 intertwines the unzip operation ue : K() K(ue()) with a renormalized

chord diagram operation ue : A() A(ue()), ue = i21/2

ue i1/2 , where i1/2 denotes

the operation of placing a factor of 1/2 on e, ue is the chord-diagram unzip operation

induced by the topological unzip, and i21/2 places factors of 1/2 on each daughter

edge. So we have Z2(ue()) = ueZ2().

This is an anomaly: if Z2 was honestly homomorphic, there should be no new chords

appearing, i.e., Z2 should intertwine unzip and its induced chord diagram operation.

Furthermore, as we prove in Chapter 3, the problem is not a shortcoming of only the

Z2 constructed here: homomorphic expansions on KTG dont exist. Our main goal

in Chapter 4 is to fix this by changing the space of KTGs slightly, and constructing a

homomorphic expansion on this new space, which we call dotted knotted trivalent graphs,

or dKTGs. We also prove that the changes dont effect the good properties we like KTGs

for.

2.3 The definition of the Kontsevich Integral

In this section we present the classical construction of the Kontsevich integral of knots,

with proofs or at least proof sketches. The extension in Chapter 3 builds on this to a

large extent, and uses the techniques explained here. The main reference we follow is

[CD], further references on the subject include [BN1] and [Ko].

Let us represent R3 as a direct product of a complex plane C with coordinate z and

a real line with coordinate t. We choose a Morse embedding for the oriented knot K: an

embedding into CR such that the coordinate t is a Morse function on K (see the figure

below).

The Kontsevich integralofK is an element in the graded completion ofA (throughout

this section, A denotes the space of chord diagrams on an oriented circle), and is defined

by the following formula:


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2

3

t1

t2

t3

t4

t

4

1

z1 z

1

DP

Z(K) =

m=0

tmin


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2.3.1 Convergence

We review the proof of the fact that each integral in the above formula is convergent. By

looking at the definition, one observes that the only way the integral may not be finite

is the (zi zi) in the denominator getting arbitrarily small near the critical points (the

boundaries of the connected components of the integration domain). This only happens

near a local minimum or maximum in the knot - otherwise the minimum distance between

strands is a lower bound for the denominator.

Roughly speaking, if a chord ck is separated from the critical value by another long

chord ck+1 ending closer to the critical value, as shown below, then the smallness in the

denominator corresponding to chord ck will be canceled by the smallness of the integration

domain for ck+1, hence the integral converges:

z

k+1

z

k

zk+1

zk

zcrit

ck+1

ck

More precisely, the integral for the long chord can be estimated as follows (using the

figures notation):

tcrittk

dzk+1 dzk+1

zk+1 zk+1

C tcrittk

d(zk+1 zk+1)

=

C|(zcrit zk) (zk+1(tcrit) zk+1(tk))| C

|zk zk|

For some constants C and C. So the integral for the long chord is as small as the

denominator for the short chord, therefore the integral converges.

Thus, the only way a divergence can occur is the case of an isolated chord, i.e. a

chord near a critical point that is not separated from it by any other chord ending. But,

by the one term relation, chord diagrams containing an isolated chord are declared to be

zero, which makes the divergence of the corresponding integral a non-issue.


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2.3.2 Invariance

Since horizontal planes cut the knot into tangles, we will use tangles and their properties

to prove the invariance of the Kontsevich integral in the class of Morse knots.

By a tangle we mean a 1-manifold embedded in [0, 1]3, whose

boundary is the union of k points on the bottom face of the cube,

positioned at

(1

k + 1,

1

2, 0), (

2

k + 1,

1

2, 0), ..., (

k

k + 1,

1

2, 0);

and l points on the top face, positioned at

(1

l + 1,

1

2, 1), (

2

l + 1,

1

2, 0),..., (

l

l + 1,

1

2, 1).

See example on the right. Two tangles are considered equal if there is an isotopy of the

cubes that fixes their boundary and takes one tangle to the other.

Tangles can be multiplied by stacking one cube on top of another and rescaling, if

the number of endpoints match.

A tangle chord diagramis a tangle supplied with a set of horizontalchords considered

up to a diffeomorphism of the tangle that preserves the horizontal fibration. Multiplica-

tion of tangles induces a multiplication of tangle chord diagrams in the obvious way.

If T is a tangle, the space AT is a vector space generated by all chord diagrams on

T, modulo the set of tangle one- and four-term relations:

The tangle one-term (or framing independence) relation is the same as the framing

independence relation for knots, i.e., asserts that a tangle chord diagram with an isolated

chord is equal to zero in AT.

For defining the tangle 4T relation, consider a tangle consisting of n parallel vertical

strands. Denote by tij the chord diagram with a single horizontal chord connecting the

i-th and j-th strands, multiplied by (1), where stands for the number of endpoints

of the chord lying on downward-oriented strands.


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... ... ...

tij = (1)

i j

The tangle 4T relation can be expressed as a commutator in terms of the tijs:

[tij + tik, tjk ] = 0,

and is illustrated in the figure below:

i i i i j j j jk k k k

+ = 0

One can check that by closing the vertical strands into a circle respecting their ori-

entations, the tangle 4T relation carries over into the ordinary 4T relation of knots.

We take the opportunity here to mention a useful lemma, which is a direct consequence

of the tangle 4T relations. A slightly different version of this appears in [BN1], and a

special case is stated in [MO].

Lemma 2.3.1. Locality. Let T be the trivial tangle consisting of n parallel vertical

strands, and D be any tangle chord diagram on T such that no chords end on the j-th

string. Let S be the sumi=j

tij in AT. Then S commutes with D in AT.

The Kontsevich integral is defined for tangles the same way it is defined for knots (by

placing a tangle in the picture instead of a knot). The advantage is that thangles can be

multiplied, and by Fubinis theorem, Z is multiplicative:

Z(T1)Z(T2) = Z(T1T2),

whenever the product T1T2 is defined.

This implies the important fact that the Kontsevich integral of the vertical con-

nected sum of knots is the connected sum of the Kontsevich integrals of the summands.

By vertical connected sum we mean that the two knots are placed directly above one


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another and connected by close parallel vertical lines. By the invariance results that

will follow this generalizes to any connected sum, and is sometimes referred to as the

factorization property, or multiplicativity, of Z.

Proposition 2.3.2. The Kontsevich integral is invariant under horizontal deformations

(deformations preserving the t coordinate, as shown in the figure below) of the knot, which

leave the levels of the critical points fixed.

Proof. Decompose the knot into a product of tangles without critical points, and other

(thin) tangles containing one unique critical point.

The following lemma, which could be considered the heart of the invariance of Z,

addresses the case of tangles without critical points. The proposition then follows from

the lemma by taking a limit. (See [CD] for more details.)

Lemma 2.3.3. LetT0 be a tangle without critical points and T a horizontal deformation

ofT0 into T1, such thatT fixes the top and the bottom of the tangle. ThenZ(T0) = Z(T1).

Proof. Let denote the differential form in the m-th term of the Kontsevich Integral:

=

P=(zi,zi)

(1)#P

(2i)mDP

mi=1

dzi dzi

zi zi

Since there are no critical points, the integration domain for any is the entire m-

simplex = {tmin < t1 < ... < tm < tmax}. Consider the product of this simplex with

the unit interval: = 0 I, and apply Stokes theorem:

=

d.

The form is exact: d = 0. The boundary of the domain is = 01+

{faces}.

To prove the lemma it is enough to show that restricted to each face is zero.


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This is the case for the faces defined by t1 = tmin or tm = Tmax, since this implies

dz1 = dz1 = 0, (or dzm = dz

m = 0), because z1 and z

1 (or zm and z

m) do not depend on

.

For the faces where tk = tk+1 for some k, the endpoints of the k-th and (k + 1)-th

chords might coincide, meaning we may not get a chord diagram at all. So, to define the

prolongation of and DP to such a face, we agree to place the k-th chord a little lower

than the (k + 1)-th chord, in the case where some of their endpoints belong to the same

string. The summands of belong to three subcases:

1) The k-th and (k + 1)-th chords connect the same two strings: we have zk = zk+1

and zk = z

k+1 or vice versa, so d(zk z

k) d(zk+1 z

k+1) = 0 and so the restriction of

to the face is zero.

2) The endpoints of the k-th and (k + 1)-th chords belong to four different strings: it

is easy to check that all choices of chords in this part of appear in mutually canceling

pairs.

3) There are three different strings containing the endpoints of the k-th and (k +1)-th

chord: a slightly more involved computation shows that this part of is indeed zero.

We will need a modification of the proof of the next lemma for the KTG case, so we

present this proof in full detail.

Proposition 2.3.4. Moving critical points. LetT0 and T1 be two tangles that differ

only in a thin needle (possibly twisted), as in the figure, such that each level {t = c}

intersects the needle in at most two points and the distance between these is , a small

positive number (it is enough that other parts of the knot do not wind through the needle).

Then Z(T0) = Z(T1).

T0 T1


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Proof. Z(T0) and Z(T1) can only differ in terms in which some chords end on the needle.

If the chord closest to the end of the needle connects the two sides of the needle (isolated

chord), then the corresponding diagram is zero by the F I (1T) relation.

So, we can assume that the one closest to the needles end is a long chord, suppose

the endpoint belonging to the needle is (zk, tk). Then, there is another choice for the

k-th chord which touches the needle at the opposite point (zk , tk), as the figure shows,

and DP will be the same for these two choices.

zk zkzk

The corresponding two terms appear in Z(T1) with opposite signs due to (1)P , and

the difference of the integrals can be estimated as follows:

tc

tk1

d(ln(zk zk))

tctk1

d(ln(zk zk))

=lnzk (tk1) zk(tk1)

zk(tk1) zk(tk1)

=

=ln1 + zk (tk1) zk(tk1)

zk(tk1) zk(tk1)

C|zk (tk1) zk(tk1)| C,where tc is the value of t at the tip of the needle, and C is a constant depending on the

minimal distance of the needle to the rest of the knot.

If the next, (k 1)-th chord is long, then the double integral corresponding to the

k-th and (k 1)-th chords is at most:

tctk2

tctk1 d(ln(z

k z

k)) tc

tk1 d(ln(z

k z

k))

d(ln(zk1 z

k1))

C tc

tk2

d(ln(zk1 zk1))

= Cln zk1(tc) zk1(tc)zk1(tk2) zk1(tk2)

CC,

where C is another constant depending on the ratio of the biggest and smallest horizontal

distance from the needle to the rest of the knot.


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If the (k 1)-th chord is short, i.e. it connects zk1 and zk1 that are both on the

needle, then we can estimate the double integral corresponding to the k-th and (k 1)-th

chords:

tctk2

tctk1

d(ln(zk zk))

tctk1

d(ln(zk zk))dzk1 dzk1

zk1 zk1

C

tctk2

(zk (tk1) zk(tk1))dzk1 dzk1|zk1 zk1|

== C

tctk2

d(zk1 zk1)

= C|zk1(tk2) zk1(tk2)| C.

Continuing to go down the needle, we see that the difference between Z(T0) and

Z(T1) in degree n is proportional to (C)n, for a constant C = max{C, C}, and by

horizontal deformations we can make tend to zero, therefore the difference tends to

zero, concluding the proof.

So far we have proved the invariance of the Kontsevich integral in the class of Morse

knots: to move critical points, one can form a sharp needle using horizontal deformations

only, then shorten or lengthen the needles arbitrarily, then deform the knot as desired

by horizontal deformations.

However, Z is not invariant under straightening humps, i. e., deformations that

change the number of critical points, as shown below. (We note that straightening the

mirror image of the hump shown is equivalent to this one, see Section 3.3 for the details.)

To fix this problem, we apply a correction factor, using the proposition below, which

is a consequence of the lemma that follows it. We sketch the proofs briefly here, see [CD]

for more details.


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Proposition 2.3.5. Let K and K be two knots differing only in a small hump in K

that is straightened in K (as in the figure). Then

Z(K

) = Z(K)Z( ).

Lemma 2.3.6. Faraway strands dont interact. Let K be a Morse knot with a

distinguished tangle T, withtbot and ttop being the minimal and maximal values of t onT.

Then, in the formula of the Kontsevich integral, for those components whose projection

on the tj axis is contained in [tbot, ttop], it is enough to consider pairings where either both

points (zj , tj ) and (z

j, tj) belong to T, or neither do.

T

Proof. (Sketch) We can shrink the tangle T into a narrow box of width , and do the

same for the rest of the knot between heights tbot and ttop. It is not hard to show that the

value of the integral corresponding to long chords (connecting the tangle to the rest of

the knot) then tends to zero.

Proof of Proposition 2.3.5. Using the notation of Lemma 2.3.6, choose T to include

just the hump, so by the assertion of the Lemma, there will be no long chords connecting

the hump to the rest of the knot in Z(K) or in Z( ). Also, in Z( ), there are no

chords above or below the hump, since the highest (resp. lowest) of those would be an

isolated chord.

Since the constant term ofZ( ) is 1, it has a reciprocal in the graded completion of

A (i.e. formal infinite series of chord diagrams). Using this we can now define an honest


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knot invariant Z by setting

Z(K) =Z(K)

Z( )c/2,

where c is the number of critical points in the Morse embedding of K that we use tocompute Z.

2.3.3 Universality

Here we state Kontsevichs theorem, and the main idea of the proof, which will apply in

the case of the extension to graphs wihout any changes. A complete, detailed proof can

be found in [CD] or [BN1], and Kontsevichs original paper [Ko].

Theorem 2.3.7. Letw be a weight system of order n. Then the a Vassiliev invariant of

order n given by the following formula has weight system w:

K w(Z(K)).

Proof. (Sketch.) Let D be a chord diagram of order n, and KD a singular knot with

chord diagram D. The theorem follows from the fact that

Z(KD) = D + {higher order terms}.

Since the denominator of Z always begins with 1 (the unit of A), it is enough to prove

that

Z(KD) = D + {higher order terms}.

Because of the factorization property and the fact that faraway strands dont interact

(Lemma 2.3.6), we can think locally. Around a single double point, we need to compute

the difference ofZ on an over-crossing and an under-crossing. These can be deformed as

follows:

Z(

) Z( ) = Z

Z

.


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Since the crossings on the bottom are now identical, by the factorization property, its

enough to consider

Z Z .Z

equals 1 (the unit of A(2), where A(2) stands for chord diagrams on two

upward oriented vertical strands), as both zi(t) and zi(t) are constant.

In Z

, the first term is 1, as always, so this will cancel out in the difference.

The next term is the chord diagram with one single chord, and this has coefficient

12i

tmaxtmin

dzdz

zz= 1, by Cauchys theorem.

So the lowest degree term of the difference is a single chord with coefficient one. Now

putting KD together, the lowest degree term in Z(KD) will be a chord diagram that has

a single chord for each double point, which is exactly D.

Note that the crucial property of Z that

Z(KD) = D + {higher order terms}

is the universality of Z, i.e., the fact that Z is an expansion.

In later chapters we will refer to the final (isotopy invariant) version of the Kontsevich

integral as Z.


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Chapter 3

The Kontsevich integral of knotted

trivalent graphs

In this chapter we build an extension of the Kontsevich integral of knots to knotted

trivalent graphs. With the exception of Theorem 3.4.1, which is from the preprint [BD1],

everything in this chapter appeared in the paper [Da].

3.1 The naive extension

We start by trying to extend the definition of the Kontsevich integral to knotted trivalent

graphs (and trivalent tangles) the natural way: consider a Morse embedding of the graph

(or tangle) in R3, as shown below. Although the graph is not planar, we understand the

framing to be the blackboard framing, in a slightly generalised sense: the normal vector

defining the framing is parallel to the plane iR R and its inner product with (i, 0)

is positive1. Define the integral by the same formula as before, requiring that t1,...,tn

are non-critical and also not the heights of vertices. (We do not do any correction or

renormalization yet.) We denote this naive extension by Z0.

1This requires that the curve is never parallel to (i, 0), but this can be achieved by a small pretur-bation for both a specific embedding and isotopies.

34


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Chapter 3. The Kontsevich integral of knotted trivalent graphs 35

2

3

t1

t2

t3

t4

t

4

1

z1 z

1

DP

Z() =

m=0

tmin


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Nice behavior under the edge delete operation.

Assuming (mistakenly) that our extension is a convergent knotted graph invariant,

consider an embedding of the graph in which the edge e to delete is a straight

vertical line, with the top vertex forming a Y, the bottom vertex resembling a

shape. (Of course such an embedding exists within each isotopy class.)

Now, if we delete the edge e, then in the result of the integral every chord diagram

in which a chord ended on e will disappear (go to zero), and the coefficient of

any other chord diagram stays unchanged (as the integral used to compute it is

unchanged). In other words, the extended Kontsevich integral commutes with theedge delete operation.

Nice behavior under edge unzip.

Let the embedding of the graph be as above. When we unzip the vertical edge e,

we do it so that the two new edges are parallel and very close to each other.

In the result of the integral, the chord diagrams that contained k chords ending on

e will be replaced by a sum of 2k chord diagrams, as each chord is replaced by the

sum of two chords, one of them ending on the first new edge, the other ending

on the second. (Since for each choice of zi on e we now have two choices.) The

coefficient for the sum of these new diagrams will be the same as the coefficient of

their parent, (since the two new edges are arbitrarily close to each other).

If we were to choose a chord to have both ends on the two new parallel edges, the

resulting integral will be zero, as zi zi will be a constant function.

Again, the coefficients of the diagrams that dont involve chords ending on e are

unchanged. Therefore, the extended Kontsevich integral, assuming it exists and is

an invariant, commutes with the unzip operation.


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3.1.2 The problem

The problem with the extension is that the integral as defined above is divergent. Causing

this are possible short chords near a vertex (i.e. those not separated from a vertex by

another chord ending). These are just like the isolated chords in the knot case but,

contrary to the knot case, we have no reason to factor out by all the chord diagrams

containing such chords. Also, if we want to drop the F I (a.k.a. 1T) relation for the sake

of working with framed graphs, we have to fix the divergence coming from the isolated

chords near critical points as well.

3.2 Eliminating the divergence

The Kontsevich Integral has been previously extended to framed links (and framed tan-

gles) by Le and Murakami in [LM], [LM2], and by Goryunov in [Go]. We use essentially

the same method as Le and Murakami, a simple version of a renormalization technique

from quantum field theory, which extends to trivalent vertices easily. In short, we know

the exact type of divergence, and thus we divide by it to get a convergent integral.

Goryunovs approach was different, using an -shift of the knot along a general framing

(not necessarily the blackboard framing).

3.2.1 The re-normalized integral Z1

We first restrict our attention to a vertex of a shape, fix a scale and chose a small

. We change the integral at the vertex by opening up the two lower strands at a

distance from the vertex, to a width at the height of the vertex.


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The old strands (solid lines on the picture) up to distance from the vertex, and

above the vertex, will be globally active, meaning that we allow any chords (long or

short) to end on them. The opening strands (dashed lines on the picture) are locally

active, meaning that we allow chords between them, but chords from outside are not

allowed to end on them. We define the value of Z1 as the limit of this new integral as

tends to zero.

We will do the same to a vertex of a Y-shape, however, we will have to restrict

our attention to these two types of vertices. (I.e. we do not allow vertices to be local

minima or maxima.) Of course, any isotopy class graph embeddings into R3 contains a

representative such that all vertices are of one of these two types, but this will cause a

problem with the invariance of Z1, which will need to be fixed later.

To get an invariant of framed graphs, we use the same method to re-normalize at the

critical points and thereby make isolated chords cause no divergence, this is why we can

drop the F I (or 1T) relation.

Proposition 3.2.1. The re-normalized integral Z1 is convergent.

Proof. It suffices to consider the case of a -shaped vertex, the other cases are similar.

Let us fix a -shaped vertex v. We will be computing the integral for the tangle (slice

of ) between a fixed level and tv, which does not contain any critical or vertex levels

other than tv, and we assume that this part of the graph all lies in one plane and all

strands except for the legs of the are vertical. As the value will turn out to be

invariant under horizontal deformations, we can do this without loss of generality, and


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since the factorization property also remains true, proving the convergence for this tangle

is enough.

If the highest chord is a long chord, the convergence is obvious. Suppose the highest

k chords, call them ci, ci1,..., cik+1, are short, as ahown in the figure below, and first

compute the integral corresponding to these.

.

.

.

t

w(t)

tv

cik+1

ci

cik

ci1ci2

tv

The globally active part corresponding to the highest short chord ci can be computed

as below.

tvti1

dzi dzi

zi zi=

tvti1

d ln(zi zi) = ln

zi(tv ) zi(tv )zi(ti1) zi(ti1)

.

The locally active part on the other hand:

tvtv

d(ln(zi zi)) = ln

zi(tv ) zi(tv )

.

The integral for this highest chord is the sum of these, and is therefore equal to:

ln

zi(ti1) zi(ti1)

= ln ln(zi(ti1) z

i(ti1)).

Let us simplify notation by calling w(t) (for width) the distance between the legs

of the at the level t. In this notation the result for the top chord is ln ln w(ti1).

For the next short chord underneath, the double integral corresponding to the two

chords is: tvti2

ln ln(w(ti1))

d ln(w(ti1)) =

=

1

2

ln ln(w(ti1))

2tvti2

=1

2

ln ln

w(ti2)

2.


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Continuing in this fashion for the k short chords, we see that the value of the integral

between the two critical levels is 1k!

ln ln

w(tik)

k, which is finite and does not

depend on .

Now suppose the next chord, cik is long. Note that this chord can only end on the

globally active part, and hence the integral corresponding to the top (k + 1) chords is

tvtik1

1

k!

ln ln w(tik)

k d(zik zik)zik zik

.

Note that since the other end of cik ends on a vertical chord, d(zik zik) =

dw(tik)2

and zik zik = C +

w(tik)

2, where C is the distance of the vertical strand from the

vertex. So we have

tvtik1

1

k!

ln ln w(tik)

k d(w(tik))w(tik) + 2C

.

We need to show that this integral is finite and converges as 0. Since the integrand

does not change sign near tv, the value (assuming it is finite) changes monotonically as

approaches 0, therefore it is enough to show that the integral is bounded by a number

that does not depend on .

We have the following obvious bounds:

1

2C+ w()

tvtik1

1

k!

ln ln w(tik)

kd(w(tik))

tvtik1

1

k!

ln ln w(tik)

k d(w(tik))w(tik) + 2C

1

2C

tv

tik1

1

k!

ln ln w(tik)

kd(w(tik)).

Thus, it is enough to show boundedness fortv

tik1

1k!

ln ln w(tik)

kd(w(tik)). Inte-

grating by parts, we find that the value of this integral is

kj=1

1

j!w(tik)

ln ln w(tik

)jtv

tik1+ [w(tik)]

tvtik1

=


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Assuming that w(tv ) = (which we can assume without loss of generality), we

obtain that the above formula equals

j=1 k

1

j!(ln ln )j

kj=1

1

j! w(tik1)(ln ln w(tik1))j

+ w(tik1).

The terms of the first sum and the lone both tend to zero as approaches zero, and

therefore the integral is bound between two different constant multiples of

k

j=1

1

j!w(tik1)(ln ln w(tik1))

j w(tik1),

and hence it converges as 0.

Showing the convergence for any further chords (short or long) is easy and is left to

the reader.

3.2.2 The good properties

Let us call the deletion (respectively, unzip) of an edge that is embedded as a vertical

line segment vertical edge delete (respectively, vertical unzip). By vertical connected sum,

we mean placing one KTG above the another and connecting them by an edge that is a

vertical line segment.

Theorem 3.2.2. Z1 is invariant under horizontal deformations that leave the critical

points and vertices fixed, and rigid motions of the critical points and vertices, explained

below. Z1 has the factorization property, and commutes with orientation switch, vertical

edge delete, edge unzip and connected sum. Moreover, it has good behavior under changing

the renormalization scale .


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By rigid motions of critical points we mean shrinking or extending a

sharp needle, like in the case of the standard Kontsevich integral (Lemma

2.3.4), with the difference that we do not allow twists on the needle, but

require the two sides of the needle to be parallel straight lines. This dif-

ference is due to dropping the framing independence relation, as adding or eliminating

twists would change the framing. We are going to study the role of framing in more detail

after proving Theorem 3.2.2. For vertices, a rigid motion is moving the vertex down two

very close edges without twists, as shown in the figure on the right.

To prove that the integral commutes with the vertical edge unzip operation and to

investigate the behavior under changing the scale , we will use the following lemma:

Lemma 3.2.3. Let w1, w2 be distinct complex numbers and let be another complex

number. Let B be the 2-strand rescaling braid defined by the map

[, T] [, T] C2

t (t, etw1, etw2).

Then

Z1(B) = expt12(T )

2i

A(2),

where t12 is the chord diagram with one chord between the two vertical strands.

Proof. The m-th term of the sum in the defining formula of Z is

1

(2i)mtm12

T

T

t1

...

T

tm1

dln(etmw1 etmw2)...dln(e

t1w1 et1w2) =

=1

(2i)mtm12

m

T

Tt1

...

Ttm1

dtmdtm1...dt1 =(t12(T ))

m

(2i)mm!,

which proves the claim.


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We note that Lemma 3.2.3 is easily extended to the case of the n-strand rescaling

braid, defined the same way, where in the statement t12 would be replaced by

tij,

though we dont use this generalization here.

We also state the following reformulation, which follows from Lemma 3.2.3 by ele-

mentary algebra:

Lemma 3.2.4. For the two strand rescaling braid B where the bottom distance between

the strands is l, and the top distance is L, shown here:l

L

,

Z1(B) = exp

ln(L/l)t12

2i ,

independently of T and .

Now we proceed to prove Theorem 3.2.2:

Proof. Factorization property. The factorization property for tangles is untouched by

the renormalization, as the height at which tangles are glued together must be non-critical

and not contain any vertices.

For the vertical connected sum of knotted graphs 1 and 2, denoted 1#2, if we con-

nect the maximum point of the 1 with the minimum of2, the minimum and maximum

renormalizations will become vertex renormalizations when computing the Kontsevich

integral of 1#2.

Invariance. To prove invariance under horizontal deformations that leave the critical

points and vertices fixed, we use the same proof as in the case of the standard integral

(Proposition 2.3.2), i.e. cut the graph into tangles with no critical points or vertices, and

thin tangles containing the vertices and critical points, apply Lemma 2.3.3 to the former

kind, then take a limit.

For invariance under rigid motions of critical points, it is enough to consider the case

of a maximum, the case of a minimum being strictly similar. Since we have proven the

invariance under horizontal deformations and the needle is not twisted, we can assume


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that the sides of the needle are two parallel lines and is the horizontal distance between

them.

Extended Retracted

By the factorization property, the value of Z1 for the

needle extended (see figure) can be written as a product of

the values for the part under the needle, the two parallel

strands, and the renormalization for the critical point that is the tip of the needle, as

shown on the right.

The value ofZ1 for the needle retracted is the product of the value for the part under

the needle and the renormalization part. What we have to show therefore is that in the

first case (needle extended) the coefficients for any diagram that contains any chords on

the parallel strands tends to zero as the width of the needle tends to zero.

This is indeed the case: the integral is 0 for any diagram on two parallel strands that

contains any short chord, since d(zk zk) = 0. For long chords, the highest long chord

can be paired up with the one ending on the other strand, as in the proof of 2.3.4. The

reason for this is that their difference commutes with any short chords that occur in the

renormalization part, by the Locality Lemma 2.3.1. Now we can use the same estimates

as in Proposition 2.3.4 to finish the proof.

To prove invariance under rigid motions of vertices, let us assume that all edges are

outgoing. All other cases are proven the same way after inserting the appropriate sign

changes. Similarly to the case of critical points, we can assume that the part we shrink

consists of two parallel strands at horizontal distance . We need to prove that the

difference of the values of Z1 for the two pictures shown below tends to zero as tends

to zero.

For the value corresponding to the left picture, just like in

the needle case, we can assume that there are no short chords

connecting the two parallel strands. The long chords ending on

the parallel strands come in pairs, with the same sign, and their coefficients are the same


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in the limit. These pairs commute with any short chords in the renormalization part by

Lemma 2.3.1. Also, by the vertex invariance relation, each sum of a pair of such chords

equals one ch

zsuzsanna dancso- on a universal finite type invariant of knotted trivalent graphs

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