marco mackaay and pedro vaz- the universal sl3-link homology

8/3/2019 Marco Mackaay and Pedro Vaz- The Universal sl3-Link Homology

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arXiv:math/0

603307v2

[math.G

T]26Jun2006

THE UNIVERSAL sl3-LINK HOMOLOGY

MARCO MACKAAY AND PEDRO VAZ

ABSTRACT. We define the universal sl3-link homology, which depends on 3

parameters, following Khovanovs approach with foams. We show that this 3-

parameter link homology, when taken with complex coefficients, can be divided

into 3 isomorphism classes. The first class is the one to which Khovanovs orig-

inal sl3-link homology belongs, the second is the one studied by Gornik in the

context of matrix factorizations and the last one is new. Following an approach

similar to Gorniks we show that this new link homology can be described in

terms of Khovanovs original sl2-link homology.

1. INTRODUCTION

In [8], following his own seminal work in [6] and Lee, Bar-Natan and Turners

subsequent contributions [11, 2, 13], Khovanov classified all possible Frobenius

systems of dimension two which give rise to link homologies via his construction

in [6] and showed that there is a universal one, given by

Z[X,a,b]/

X2 aX b

.

Working over C, one can take a and b to be complex numbers and study the cor-responding homology with coefficients in C. We refer to the latter as the sl2-link

homologies over C, because they are all related to the Lie algebra sl2 (see [8]).Using the ideas in [8, 11, 13], it was shown in [12] that there are only two isomor-

phism classes of sl2-link homologies over C. Given a, b C, the isomorphismclass of the corresponding link homology is completely determined by the number

of distinct roots of the polynomial X2 aX b. The original Khovanov sl2-linkhomology KH(L,C) corresponds to the choice a = b = 0.

Bar-Natan [2] obtained the universal sl2-link homology in a different way, us-ing a clever setup with cobordisms modulo relations. He shows how Khovanovs

original construction of the sl2-link homology [6] can be used to define a universalfunctorU from the category of links, with link cobordisms modulo ambient isotopyas morphisms, to the homotopy category of complexes with values in the category

of1 + 1-dimensional cobordisms modulo a finite set of universal relations. In the

same paper he introduces the tautological homology construction, which producesan honest homology theory from U. To obtain a finite dimensional homology onehas to impose the extra relations

= a + b and = + a

on the cobordisms.1
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2 MARCO MACKAAY AND PEDRO VAZ

In [7] Khovanov showed how to construct a link homology related to the Lie

algebra sl3. Instead of 1 + 1-dimensional cobordisms, he uses webs and singular

cobordisms modulo a finite set of relations, one of which is X3

= 0. Gornik [4]studied the case when X3 = 1, which is the analogue of Lees work for sl3. Tobe precise, Gornik studied a deformation of the Khovanov-Rozansky theory [9] for

sln, where n is arbitrary. Khovanov and Rozansky followed a different approachto link homology using matrix factorizations which conjecturally yields the same

for sl3 as Khovanovs approach using singular cobordisms modulo relations [7].However, in this paper we restrict to n = 3 and only consider Gorniks results forthis case.

In the first part of this paper we construct the universal sl3-link homology overZ[a,b,c]. For this universal construction we rely heavily on Bar-Natans [2] workon the universal sl2-link homology and Khovanovs [7] work on his original sl3-link homology. We first impose a finite set of relations on the category of webs and

foams, analogous to Khovanovs [7] relations for his sl3-link homology. We showthat these relations imply certain identities between foams which are defined over

Z and are analogous to Bar-Natans universal relations for sl2 in [2]. The latter aresufficient to obtain a chain complex for each link which is invariant under the Rei-

demeister moves up to homotopy. However, they are insufficient for extending our

construction to a functor, defined up to 1, from the category of links to the homo-topy category of complexes, for which we need the full set of relations including

the ones depending on a, b and c. 1 This is the major difference with Bar-Natansapproach to the universal sl2-homology. To obtain a finite-dimensional homologyfrom our complex we use the tautological homology construction like Khovanov

did in [7] (the name tautological homology was coined by Bar-Natan in [2]). We

denote this universal sl3-homology by Ua,b,c(L), which by the previous results is

an invariant of the linkL.In the second part of this paper we work over C and take a,b,c to be complex

numbers, rather than formal parameters. We show that there are three isomorphism

classes ofUa,b,c(L,C), depending on the number of distinct roots of the polynomial

f(X) = X3 aX2 bX c, and study them in detail. Iff(X) has only one root,with multiplicity three of course, then Ua,b,c(L,C) is isomorphic to Khovanovs

original sl3-link homology, which in our notation corresponds to U0,0,0(L,C). If

f(X) has three distinct roots, then Ua,b,c(L,C) is isomorphic to Gorniks sl3-link

homology [4], which corresponds to U0,0,1(L,C). The case in which f(X) has twodistinct roots, one of which has multiplicity two, is new and had not been studied

before to our knowledge, although in [3] and [5] the authors make conjectures

which are compatible with our results. We prove that there is a degree-preservingisomorphism

Ua,b,c(L,C)=

LL

KHj(L)(L,C),

1We thank M. Khovanov for spotting this problem in an earlier version of our paper.


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THE UNIVERSAL sl3-LINK HOMOLOGY 3

where j(L) is a shift of degree 2lk(L, L\L). This isomorphism does not takeinto account the internal grading of the Khovanov homology.

We have tried to make the paper reasonably self-contained, but we do assumefamiliarity with the papers by Bar-Natan [1, 2], Gornik [4] and Khovanov [6, 7, 8].

2. THE UNIVERSAL sl3-LINK HOMOLOGY

Let L be a link in S3 and D a diagram of L. In [7] Khovanov constructed ahomological link invariant associated to sl3. The construction starts by resolvingeach crossing ofD in two different ways, as in figure 1. A diagram obtained by

0

vvnnnnn

nnnn

nn 1

((PPPP

PPPP

PPP

1

hhPPPPPPPPPPP 066nnnnnnnnnnn

FIGURE 1. 0 and 1 resolutions of crossings

resolving all crossings of D is an example of a web. A web is a trivalent planargraph where near each vertex all the edges are oriented in or out (see figure 2).

We also allow webs without vertices, which are oriented loops. Note that by defini-

tion our webs are closed; there are no vertices with fewer than 3 edges. Whenever

FIGURE 2. In and out orientations near a vertex

it is necessary to keep track of crossings after their resolution we mark the corre-

sponding edges as in figure 3. A foam is a cobordism with singular arcs between

*

FIGURE 3. Marked edges corresponding to a crossing in D

two webs. A singular arc in a foam f is the set of points off that have a neighbor-hood homeomorphic to the letter Y times an interval (see the examples in figure 4).

Interpreted as morphisms, we read foams from bottom to top by convention, and

the orientation of the singular arcs is by convention as in figure 4. Foams can have

dots that can move freely on the facet to which they belong, but are not allowed to

cross singular arcs. Let Z[a,b,c] be the ring of polynomials in a,b,c with integercoefficients.


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2.1. Universal local relations. In order to construct the universal theory we divide

Foam by the local relations = (3D,CN,S, ) below.

= a + b + c (3D)

= + + a

+

b (CN)

= = 0, = 1 (S)

Recall from [7] that theta-foams are obtained by gluing three ori-ented disks along their boundaries (their orientations must coincide),

as shown on the right. Note the orientation of the singular circle. Let

, , denote the number of dots on each facet. The () relation says that for , or 2

( , , ) =

1 ( , , ) = (1, 2, 0) or a cyclic permutation1 ( , , ) = (2, 1, 0) or a cyclic permutation0 else

()

Reversing the orientation of the singular circle reverses the sign of (,,). Note

that when we have three or more dots on a facet of a foam we can use the (3D)relation to reduce to the case where it has less than three dots.

A closed foam f can be viewed as a morphism from the empty web to itselfwhich by the relations (3D, CN, S, ) is an element ofZ[a,b,c]. It can be checked,as in [7], that this set of relations is consistent and determines uniquely the eval-

uation of every closed foam f, denoted C(f). Define a q-grading on Z[a,b,c] asq(1) = 0, q(a) = 2, q(b) = 4 and q(c) = 6. As in [7] we define the q-grading of afoam f with d dots by

q(f) = 2(f) + (f) + 2d,

where denotes the Euler characteristic.

Definition 2.2. Foam/ is the quotient of the categoryFoam by the local relations

. For webs , and for families fi HomFoam/(, ) and ci Z[a,b,c]

we imposei cifi = 0 if and only if

i ciC(g

fig) = 0 holds, for all g HomFoam/(, ) and g

HomFoam/(, ).


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Lemma 2.3. We have the following relations in Foam/:

+ = + (4C)

= (RD)

= (DR)

= (SqR)

Proof. Relations (4C) and (RD) are immediate and follow from (CN) and ().Relations (DR) and (SqR) are proved as in [7] (see also lemma 2.7)

The following equality, and similar versions, which corresponds to an isotopy,

we will often use in the sequel

(1) =

where denotes composition of foams.In figure 5 we also have a set of useful identities which establish the way we can

exchange dots between faces. These identities can be used for the simplification of

foams and are an immediate consequence of the relations in .

2.2. Invariance under the Reidemeister moves. In this subsection we prove in-

variance of under the Reidemeister moves. The proof only uses the relationswhich are defined over Z. The main result of this section is the following

Theorem 2.4. D is invariant under the Reidemeister moves up to homotopy, inother words it is an invariant in Kom/h(Foam/).

Proof. To prove invariance under the Reidemeister moves we work diagrammati-

cally and only use the identities in lemma 2.3 along with the (S) relation.


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+ + = a

+ + = b

= c

FIGURE 5. Exchanging dots between faces. The relations are the

same regardless of which edges are marked and the orientation on

the singular arcs.

Reidemeister I. Consider diagrams D and D that differ only in a circular regionas in the figure below.

D = D =

We give the homotopy between complexes D and D in figure 6. It is imme-

D :

D :

g0 =

d =

//

h = +

oo

0 //

f0 =

OO

0

0

OO

FIGURE 6. Invariance under Reidemeister I

diate that g0f0 = Id( ). To see that df0 = 0 use (DR) near the top of df0 andthen (RD). The equality dh = id(

) follows from (DR) (note the orientations onthe singular circles) and f0g0 + hd = id( ) follows from (4C). Therefore D ishomotopy-equivalent to D.


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Reidemeister IIa. Consider diagrams D and D that differ in a circular region, asin the figure below.

D = D =

We leave to the reader the task of checking that the diagram in figure 7 defines

a homotopy between the complexes D and D:

g and f are morphisms of complexes (use only isotopies); g1f1 = IdD1 (use (RD));

f0g0 + hd = IdD0 and f2g2 + dh = IdD2 (use isotopies);

f1g1 + dh + hd = IdD1 (use (DR)).

Reidemeister IIb. Consider diagrams D and D that differ only in a circular region,as in the figure below.

D = D =

Again, checking that the diagram in figure 8 defines a homotopy between the com-

plexes D and D is left to the reader:

g and f are morphisms of complexes (use only isotopies); g1f1 = IdD1 (use (RD) and (S));

f0g0 + hd = IdD0 and f2g2 + dh = IdD2 (use (RD) and (DR));

f1g1 + dh + hd = IdD1 (use (DR), (RD), (4C) and (SqR)).

D :

D :

0

0 0

I

fg0

FIGURE 7. Invariance under Reidemeister I Ia


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D :

D : 0 0

0

f0

FIGURE 8. Invariance under Reidemeister IIb

Reidemeister III. Consider diagrams D and D that differ only in a circular region,as in the figure below.

D = D =

We prove that D is homotopy equivalent to D by showing that both com-plexes are homotopy equivalent to a third complex denoted Q (the bottom com-plex in figure 9). Figure 9 shows that D is homotopy equivalent to Q. Byapplying a symmetry relative to a horizontal axis crossing each diagram in Dwe obtain a homotopy equivalence between D and Q. It follows that D ishomotopy equivalent to D.

Theorem 2.4 allows us to use any diagram D of L to obtain the invariant inKom/h(Foam/) and justifies the notation L for D.

2.3. Functoriality. It is clear that the construction and the results of the previ-

ous sections can be extended to the category of tangles, following Bar-Natans

approach in [2]. One can then prove functoriality of as Bar-Natan does. Theproofs of lemmas 8.6-8.8 in [2] are identical. The proof of lemma 8.9 follows the


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D

:

:

Q

I

I

I

I

I

I

FIGURE 9. First step of invariance under Reidemeister III. Acircle attached to the tail of an arrow indicates that the correspond-

ing morphism has a minus sign.

same reasoning but uses the homotopies of our subsection 2.2. Without giving any

details of this generalization, we state the main result. Let Kom/h(Foam/)denote the category Kom/h(Foam/) modded out by 1. Then

Proposition 2.5. defines a functorLink Kom/h(Foam/).


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Let be a resolution of a link L and let E() be the set of all edges in . In [7]Khovanov defines the following algebra (in his case for a = b = c = 0).

Definition 3.3. Let R() be the free commutative algebra generated by the ele-ments Xi, with i E(), modulo the relations

(2) Xi + Xj + Xk = a, XiXj + XjXk + XiXk = b, XiXjXk = c,

for any triple of edges i,j,k which share a trivalent vertex.

The following definitions and results are analogous to Gorniks results in sec-

tions 2 and 3 of [4]. Let S = { , , }.

Definition 3.4. A coloring of is defined to be a map : E() S. Denote theset of all colorings by S(). An admissible coloring is a coloring such that

(3)

a = (i) + (j) + (k)b = (i)(j) + (j)(k) + (i)(k)

c = (i)(j)(k),

for any edges i,j,k incident to the same trivalent vertex. Denote the set of alladmissible colorings by AS().

Of course admissibility is equivalent to requiring that the three colors (i), (j)and (k) be all distinct.

A simple calculation shows that f(Xi) = 0 in R(), for any i E(). There-fore, for any edge i E(), there exists a homomorphism of algebras fromC[X]/(F(X)) to R() defined by X Xi. Thus, for any coloring , we de-fine

Definition 3.5.

Q() =iE()

Q(i)(Xi) R().

Lemma 3.2 implies the following corollary.

Corollary 3.6. S()

Q() = 1,

Q()Q() = Q,

where is the Kronecker delta.

Note that the definition ofQ() implies that

(4) XiQ() = (i)Q().The following lemma is our analogue of Gorniks theorem 3.

Lemma 3.7. For any non-admissible coloring , we have

Q() = 0.

For any admissible coloring , we have

Q()R() = C.


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Therefore, we get a direct sum decomposition

R() = AS()

CQ().

Proof. Let be any coloring and let i,j,k E() be three edges sharing a triva-lent vertex. By the relations in (2) and equation (4), we get

(5)

aQ() = ((i) + (j) + (k))Q()bQ() = ((i)(j) + (j)(k) + (i)(k))Q()cQ() = (i)(j)(k)Q().

If is non-admissible, then, by comparing (3) and (5), we see that Q() vanishes.Now suppose is admissible. Recall that R() is a quotient of the algebra

(6) iE()

C[Xi]/ (f(Xi)) .

Just as in definition 3.5 we can define the idempotents in the algebra in (6), which

we also denote Q(). By the Chinese Remainder Theorem, there is a projectionof the algebra in (6) onto C, which maps Q() to 1 and Q() to 0, for any = . It is not hard to see that, since is admissible, that projection factorsthrough the quotient R(), which implies the second claim in the lemma.

As in [7], the relations in figure 5 show that R() acts on C() by the usual actioninduced by the cobordism which merges a circle and the relevant edge of . Letus write C() = Q()C(). By corollary 3.6 and lemma 3.7, we have a directsum decomposition

(7) C() =AS()

C().

Note that we have

(8) z C() i, Xiz = (i)z

for any AS().Let be a coloring of the arcs ofL by , and . Note that induces a unique

coloring of the unmarked edges of any resolution ofL.

Definition 3.8. We say that a coloring of the arcs of L is admissible if there ex-ists a resolution ofL which admits a compatible admissible coloring. Note that ifsuch a resolution exists, its coloring is uniquely determined by , so we use thesame notation. Note also that an admissible coloring of induces a unique admis-sible coloring ofL. If is an admissible coloring, we call the elements in C()admissible cochains. We denote the set of all admissible colorings of L by AS(L).

We say that an admissible coloring ofL is a canonical coloring if the arcs be-longing to the same component of L have the same color. If is a canonicalcoloring, we call the elements in C() canonical cochains. We denote the set ofcanonical colorings ofL by S(L).


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Note that, for a fixed AS(L), the admissible cochain groups C() form asubcomplex Ca,b,c(L,C) C

a,b,c(L,C) whose homology we denote by U

a,b,c(L,C).

The following lemma shows that only the canonical cochain groups matter, asGornik indicated in his remarks before his main theorem 2 in [4].

Theorem 3.9.

Ua,b,c(L,C) =S(L)

Ua,b,c(L,C).

Proof. By (7) we have

Ua,b,c(L,C) =AS(L)

Ua,b,c(L,C).

Let us now show that Ua,b,c(L,C) = 0 if is admissible but non-canonical. Let

1 2

5

3 4

* 1 2

FIGURE 10. Ordering edges

and be the diagrams in figure 10, which are the boundary of the cobordismwhich defines the differential in Ca,b,c(L,C), and order their edges as indicated.Up to permutation, the only admissible colorings of are

* and

*

1 2

.

Up to permutation, the only admissible colorings of are

and

0 1

.

Note that only 2 and 0 can be canonical. We get

(9)

0 C0

(),

C1()= C

1(),

C2() 0.


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Note that the elementary cobordism has to map colorings to compatible colorings.

This explains the first and the third line. Let us explain the second line. Apply the

elementary cobordisms

and use relation (RD) on page 6 to obtainthe linear map C1,2

() C1,2

() given by

z ( )z.

Since = , we see that this map is injective. Therefore the map C1,2

()

C1,2() is injective too. A similar argument, using the (DR) relation, shows thatC1,2() C1,2(

) is injective. Therefore both maps are isomorphisms.

Next, let be admissible but non-canonical. Then there exists at least one cross-ing, denoted c, in L which has a resolution with a non-canonical coloring. LetCa,b,c(L

1,C) be the subcomplex of C

a,b,c(L,C) defined by the resolutions of L

in which c has been resolved by the 1-resolution. Let Ca,b,c(L0,C) be the com-

plex obtained from Ca,b,c(L,C) by deleting all resolutions which do not belongto Ca,b,c(L

1,C) and all arrows which have a source or target which is not one of

the remaining resolutions. Note that we have a short exact sequence of complexes

(10) 0 Ca,b,c(L1,C) C

a,b,c(L,C) C

a,b,c(L

0,C) 0.

The isomorphism in (9) shows that the natural map

Ca,b,c(L0,C) C

+1a,b,c(L

1,C),

defined by the elementary cobordisms which induce the connecting homomor-

phism in the long exact sequence associated to (10), is an isomorphism. By ex-

actness of this long exact sequence we see that U

a,b,c

(L,C) = 0.

Lemma 3.10. For any AS(), we have C() = C.

Proof. We use induction with respect to v, the number of trivalent vertices in .The claim is obviously true for a circle. Suppose has a digon, with the edgesordered as in figure 11. Note that X1 = X4 R() holds as a consequence

1

2 3

4

*

*

1

FIGURE 11. Ordering edges in digon

of the relations in (2). Let be the web obtained by removing the digon, as infigure 11. Up to permutation, the only possible admissible colorings of and the


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corresponding admissible coloring of are

*

*

*

*

1 2

.

The Digon Removal isomorphism in lemma 2.7 yields

C1() C2()= C(

) C().

By induction, we have C() = C, so dim C1() + dim C2() = 2. For

symmetry reasons this implies that dim C1() = dim C2() = 1, which provesthe claim. To be a bit more precise, let B, and B, be the following two coloredcobordisms:

B, =

, B, =

.

Note that we have

B,+ B, = 0

by

= 0,

and by

=

we have

B,+ B, = idC ().

These two identities imply

( )B, = ( )B, = idC ().

Therefore we conclude that C1() and C2() are non-zero, which for dimen-sional reasons implies dim C1() = dim C2() = 1.

Now, suppose contains a square, with the edges ordered as in figure 12 left.Let and be the two corresponding webs under the Square Removal isomor-phism in lemma 2.7. Up to permutation there is only one admissible non-canonical

coloring and one canonical coloring:

* *

* *

canonical non-canonical

.


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1

2 3

4

7

8

6

5 ** 1 3

1

2

FIGURE 12. Ordering edges in square

Let us first consider the canonical coloring. Clearly C() is isomorphic toC(

), where is the unique compatible canonical coloring, because there isno compatible coloring of. Therefore the result follows by induction.

Now consider the admissible non-canonical coloring. As proved in theorem 3.9

we have the following isomorphism:

* *

=

* .

By induction the right-hand side is one-dimensional, which proves the claim.

Thus we arrive at Gorniks main theorem 2. Note that there are 3n canonicalcolorings of L, where n is the number of components of L. Note also that thehomological degrees of the canonical cocycles are easy to compute, because we

know that the canonical cocycles corresponding to the oriented resolution without

vertices have homological degree zero.

Theorem 3.11. The dimension ofUa,b,c(L,C) equals 3n, where n is the number of

components ofL.For any S(L), there exists a non-zero elementa U

ia,b,c(L,C), unique up

to a scalar, where

i =

(1,2)SS, 1=2

lk(1(1), 1(2)).

3.2. Two distinct roots. In this section we assume that f(X) = (X)2(X),with = . We follow an approach similar to the one in the previous section. Firstwe define the relevant idempotents. By the Chinese Remainder Theorem we have

C[X]/(f(X)) = C[X]/((X )2) C[X]/(X ).

Definition 3.12. Let Q and Q be the idempotents in C[X]/(f(X)) correspond-ing to (1, 0) and (0, 1) in C[X]/((X )2) C[X]/(X ) under the aboveisomorphism.

Again it is easy to compute the idempotents explicitly:

Q = 1 (X )2

( )2, Q=

(X )2

( )2.


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By definition we get

Lemma 3.13.

Q + Q= 1,QQ= 0,

Q2 = Q, Q2= Q.

Throughout this subsection let S = {, }. We define colorings of webs andadmissibility as in definition 3.4. Note that a coloring is admissible if and only if

at each trivalent vertex the, unordered, incident edges are colored ,,. Let bea web and a coloring. The definition of the idempotents Q() in R() is thesame as in definition 3.5. Clearly, corollary 3.6 also holds in this section. However,

equation 4 changes. By the Chinese Remainder Theorem, we get

(11) (Xi )Q() = 0, if (i) = (Xi )2Q() = 0, if (i) = .

Lemma 3.7 also changes. Its analogue becomes:

Lemma 3.14. For any non-admissible coloring , we have

Q() = 0.

Therefore, we have a direct sum decomposition

R() =AS()

Q()R().

For any AS() , we have dim Q()R() = 2m , where m is the number

of cycles in 1() .

Proof. First we prove that inadmissible colorings yield trivial idempotents. Let be any coloring of and let i,j,k be three edges sharing a trivalent vertex. Firstsuppose that all edges are colored by . By equations (11) we get

aQ() = (Xi + Xj + Xk)Q() = 3Q(),

which implies that Q() = 0, because a = 2 + and = .Next suppose (i) = (j) = and (k) = . Then

aQ() = (Xi + Xj + Xk)Q() = (2+ Xk)Q().

Thus XkQ() = (2 )Q(). Therefore we get

0 = (Xk )2Q() = ( )

2Q(),

which again implies that Q() = 0.Finally, suppose i,j,k are all colored by . Then we have

(Xi )2 + (Xj )

2 + (Xk )2

Q() = 0.

Using the relations in (2) we get

(Xi )2 + (Xj )

2 + (Xk )2 = ( )2,

so we see that Q() = 0.


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Now, let be an admissible coloring. Note that the admissibility conditionimplies that 1() consists of a disjoint union of cycles. To avoid confusion, let

us remark that we do not take into consideration the orientation of the edges whenwe speak about cycles, as one would in algebraic topology. What we mean by a

cycle is simply a piece-wise linear closed loop. Recall that R() is a quotient ofthe algebra

(12)iE()

C[Xi]/ (f(Xi))

and that we can define idempotents, also denoted Q(), in the latter. Note that bythe Chinese Remainder Theorem there exists a homomorphism of algebras which

projects the algebra in (12) onto

(13) (i)=

C[Xi]/ (Xi )2

(i)=

C[Xi]/ (Xi ) ,

which maps Q() to 1 and Q() to 0, for any = . Define R() to be thequotient of the algebra in (13) by the relations Xi+ Xj = 2, for all edges i and jwhich share a trivalent vertex and satisfy (i) = (j) = . Note that XiXj =

2

also holds in R(), for such edges i and j.Suppose that the edges i,j,k are incident to a trivalent vertex in and that they

are colored ,,. It is easy to see that by the projection onto R() we get

Xi + Xj + Xk aXiXj + XiXk + XjXk bXiXjXk c.

Therefore the projection descends to a projection from R() onto R(). Since

Q() is mapped to 1 and Q() to 0, for all = , we see that the projectionrestricts to a surjection of algebras

Q()R() R().

A simple computation shows that the equality

Xi + XjQ() = 2Q()

holds in R(), which implies that the surjection above is an isomorphism of alge-bras. This proves the final claim in the lemma.

As in (8), for any AS(), we get

(14) z C()

(Xi )z = 0, i : (i) = ,(Xi )2z = 0, i : (i) = .

Let be a coloring of the arcs of L by and . Note that induces a uniquecoloring of the unmarked edges of any resolution ofL. We define admissible andcanonical colorings ofL as in definition 3.8.

Note, as before, that, for a fixed admissible coloring of L, the admissiblecochain groups C() form a subcomplex C

a,b,c(L,C) C

a,b,c(L,C) whose


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homology we denote by Ua,b,c(L,C). The following theorem is the analogue oftheorem 3.9.

Theorem 3.15.

Ua,b,c(L,C) =S(L)

Ua,b,c(L,C).

Proof. By lemma 3.14 we get

Ua,b,c(L,C) =AS(L)

Ua,b,c(L,C).

Let us now show that Ua,b,c(L,C) = 0 if is admissible but non-canonical.

Let and be the diagrams in figure 10, which are the boundary of the cobordismwhich induces the differential in Ca,b,c(L,C), and order their edges as indicated.

The only admissible colorings of are

*

*

*

*

*

1 2 3 4 5

.

The only admissible colorings of are

0 1

2

5

.

Note that only 3, 4, 5, 0 and

5 can be canonical. We get

(15)

0 C0

(),

C1()= C

1(),

C2()= C

2(),

C3() 0,C4() 0,C5() C5(

).

Note that the last line in the list above only states that the cobordism induces a map

from one side to the other or vice-versa, but not that it is an isomorphism in general.

The second and third line contain isomorphisms. Let us explain the second line,

the third being similar. Apply the elementary cobordism and userelation (RD) on page 6 to obtain the linear map C

1,2() C

1,2() given by

z ( X1)z.


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Suppose (X1 )z = 0. Then z C0

(), because X1z = X2z = z . This

implies that z C1,2

() C0

() = {0}. Thus the map above is injective, and

therefore the map C1,2() C1,2() is injective. A similar argument, usingthe relation (DR), shows that C1,2() C1,2(

) is injective. Therefore both

maps are isomorphisms.

The isomorphisms in (15) imply that Ua,b,c(L,C) = 0 holds, when is admis-sible but non-canonical, as we explained in the proof of theorem 3.9.

Let C() be a canonical cochain group. In this case it does not suffice to com-pute the dimensions ofC(), for all and , because we also need to determinethe differentials. Therefore we first define a canonical cobordism in C().

Definition 3.16. Let S(L). We define a cobordism (): by gluingtogether the elementary cobordisms in figure 13 and multiplying by Q(). Wecall () the canonical cobordism in C().

* * :

* * :

if belong to different -cycles

if belong to same -cycle

FIGURE 13. Elementary cobordisms

For any canonically colored web, we can find a way to build up the canonical

cobordism using only the above elementary cobordisms with canonical colorings,

except when we have several digons as in figure 14 where we might have to stick in

two digons at a time to avoid getting webs with admissible non-canonical colorings.

FIGURE 14. several digons

There is a slight ambiguity in the rules above. At some point we may have

several choices which yield different cobordisms, depending on the order in which


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we build them up. To remove this ambiguity we order all arcs of the link, which

induces an ordering of all unmarked edges in any of its resolutions, By convention

we first build up the square or digon which contains the lowest order edge.With these two observations in mind, it is not hard to see that the rules defining

() are consistent. One can check that the two different ways of defining it forthe square-digon webs in figure 15 yield the same cobordism indeed.

* * * *

FIGURE 15. Square-digons with coloring

Recall that the definition ofR(), which appears in the following lemma, canbe found in the proof of lemma 3.14, where we showed that R() = Q()R().

Lemma 3.17. C() is a free cyclic R()-module generated by (), for any S().

Proof. We use induction with respect to v, the number of trivalent vertices in .The claim is obviously true for a circle. Suppose has a digon, with the edgesordered as in figure 11. Note that X1 = X4 R() holds as a consequence of therelations in (2). Let be the web obtained by removing the digon, as in lemma 2.7.The possible canonical colorings of and the corresponding canonical coloringsof are

*

*

*

*

*

*

1 2 3 1

2

3

.

We treat the case of1 first. Since the Digon Removal isomorphism in lemma 2.7commutes with the action ofX1 = X4, we see that

C1()= C

1() C

1().

By induction C1

() is a free cyclic R1

()-module generated by 1

(). Note

that the isomorphism maps

1(), 0

and

0,

1()

to

1() and X21().

Since dim R1() = 2dim R1(), we conclude that C1() is a free cyclic

R1()-module generated by 1().


24/29


Now, let us consider the case of 2 and 3. The Digon Removal isomorphism inlemma 2.7 yields

C2() C3() = C2() C3().

Note that 2 = 3 holds and by induction C2(

) = C3

() is a free cyclic

R2

() = R3

()-module. As in the previous case, by definition of the canonicalgenerators, it is easy to see that the isomorphism maps

R2

()2

() R3

()3

()

to

R2()2() R3()3().

Counting the dimensions on both sides of the isomorphism, we see that this proves

the claim in the lemma for C2() and C3().We could also have

* ,

but the same arguments as above apply to this case.

If we have several digons as in figure 14, similar arguments prove the claim

when we stick in two digons at a time.

Next, suppose contains a square, with the edges ordered as in figure 12 left.Let and be the two corresponding webs under the Square Removal isomor-phism in lemma 2.7. There is a number of possible canonical colorings. Note that

there is no canonical coloring which colors all external edges by . To prove theclaim for all canonical colorings it suffices to consider only two: the one in which

all external edges are colored by and the coloring

**

All other cases are similar. Suppose that all external edges are colored by , thenthere are two admissible colorings:

* * and

* *

1 2

.

Note that only 2 can be canonical. Clearly there are unique canonical coloringsof and , with both edges colored by , which we denote and . Theisomorphism yields

C1() C2()= C(

) C().


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Suppose that the two edges in belong to the same -cycle. We denote the numberof -cycles in by m. Note that the number of-cycles in equals m + 1.

By induction C

(

) = R

(

)

(

) and C

(

) = R

(

)

(

) arefree cyclic modules of dimensions 2m and 2m+1 respectively. Since 1 is non-canonical, we know that C1()

= C(), using the isomorphisms in (15) and

the results above about digon-webs. Therefore, we see that dim C1() = 2m and

dim C2() = 2m+1. By construction, we have

() (, 2()) .

The isomorphism commutes with the actions on the external edges and R2() isisomorphic to R(

), so we get

R2()2()= R(

)().

For dimensional reasons, this implies that C2() is a free cyclic R2()-module

generated by 2().Now suppose that the two edges in belong to different -cycles. This timewe denote the number of-cycles in and by 2m+1 and 2m respectively. Westill have C1()

= C(), so, by induction, we have dim C1() = 2

m+1 and,

therefore, C2() = 2m. Consider the intermediate web colored by as

indicated and the map between and in figure 16. By induction, C() is

* *//

*

FIGURE 16

a free cyclic R()-module generated by (

). By construction, we seethat 2() is mapped to

X6() X1(

),

which is non-zero. Similarly we see that X12() is mapped to

X1X6() X21(

) =X1X6(

) (2X1 2)(

).

The latter is also non-zero and linearly independent from the first element. Sincethe map clearly commutes with the action of all elements not belonging to edges

in the -cycle ofX1, the above shows that, for any nonzero element Z R2(),the image ofZ2() in C(

) is non-zero. Therefore, we see that

dim R2()2() = dim R2() = 2m.

For dimensional reasons we conclude that Q2()C() is a free cyclic R2()-module generated by 2().


26/29


Finally, let us consider the canonical coloring

** .

In this case C() is isomorphic to C(), where is the unique compatible

canonical coloring of , because there is no compatible coloring of . Note thatR() is isomorphic to R(

) and () is mapped to (). Therefore the

result follows by induction.

Finally we arrive at our main theorem in this subsection.

Theorem 3.18.

Uia,b,c(L,C)= L

L

KHij(L),(L,C),

where j(L) = 2lk(L, L\L).

Proof. By theorem 3.15 we know that

Uia,b,c(L,C) =S(L)

Uia,b,c(L,C).

Let S(L) be fixed and let L be the sublink ofL consisting of the componentscolored by . We claim that

(16) Uia,b,c(L,C)= KHij(L

),(L,C),

from which the theorem follows. First note that, without loss of generality, we may

assume that = 0, because we can always apply the isomorphism

.

Let C() be a canonical cochain group. By lemma 3.17, we know that C() isa free cyclic R()-module generated by (). Therefore we can identify anyX R() with X(). There exist isomorphisms

(17) R() = C[Xi | (i) = ]/

Xi + Xj , X2i

= Am,

where A = C[X]/

X2

. As before, the relations Xi + Xj = 0 hold wheneverthe edges i and j share a common trivalent vertex and m is the number of-cyclesin . Note also that XiXj = 0 holds, ifi and j share a trivalent vertex. The firstisomorphism in (17) is immediate, but for the definition of the second isomorphism

we have to make some choices. First of all we have to chose an ordering of the arcs

of L. This ordering induces a unique ordering on all the unmarked edges of ,where we use Bar-Natans [1] convention that, if an edge in is the fusion of twoarcs ofL, we assign to that edge the smallest of the two numbers. Now delete alledges colored by . Consider a fixed -cycle. In this -cycle pick the edge i withthe smallest number in our ordering. This edge has an orientation induced by the

orientation ofL.We identify the -cycle with a circle, by deleting all vertices in the -cycle,

oriented according to the orientation of the edge i. If the circle is oriented clock-wise we say that it is negatively oriented, otherwise we say that it is positively


27/29


oriented. The circles corresponding to the -cycles are ordered according to theorder of their minimal edges. They can be nested. As in Lees paper [11] we say

that a circle is positively nested if any ray from that circle to infinity crosses theother circles in an even number of points, otherwise we say that it is negatively

nested. The isomorphism in (17) is now defined as follows. Given the r-th -cyclewith minimal edge i we define

Xi 1 X 1,

where X appears as the r-th tensor factor. If the orientation and the nesting ofthe -cycle have the same sign, then = +1, and if the signs are opposite, then = 1. The final result, i.e. the claim of this theorem, holds true no matter whichordering of the arcs ofL we begin with. It is easy to work out the behaviour of thecanonical generators with respect to the elementary cobordisms as can be seen in

figure 17. For the cobordisms shown in figure 17 having one or more cycles there

is also a version with one cycle inside the other cycle or a cycle inside a digon.The two bottom maps in figure 17 require some explanation. Both can only be

understood by considering all possible closures of the bottoms and sides of their

sources and targets. Since, by definition, the canonical generators are constructed

step by step introducing the vertices of the webs in some order, we can assume,

without loss of generality, that the first vertices in this construction are the ones

shown. With this assumption the open webs at the top and bottom of the cobor-

disms in the figure are to be closed only by simple curves, without vertices, and

the closures of these cobordisms, outside the bits which are shown, only use cups

and identity cobordisms. Baring this in mind, the claim implicit in the first map is

a consequence of relation 1.

For the second map, recall that belong to different -cycles. Therefore there

are two different ways to close the webs in the target and source: two cycles side-by-side or one cycle inside another cycle. We notice that from theorem 3.15 we

have the isomorphism

* * =

.

We apply this isomorphism to the composite of the source foam and the elementary

foam and to the target foam of the last map in figure 17. Finally use equation 1 and

relation (CN) to the former to see that both foams are isotopic.

Note that in Ca,b,c(L,C) we only have to consider elementary cobordisms atcrossings between two strands which are both colored by . With the identifica-tion of

R()and

A

m as above, it is now easy to see that the differentials in

Ca,b,c(L,C) behave exactly as in Khovanovs original sl2-theory for L.The degree of the isomorphism in (16) is easily computed using the fact that in

both theories the oriented resolution has homological degree zero. Therefore we

get an isomorphism

Uia,b,c(L,C)= KHij(L

),(L,C).


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, -

= -

, -

belong to same

-cycle in

* *

+

belong to different

-cycles in

* *

FIGURE 17. Behaviour of canonical generators under elementary cobordisms

Acknowledgements We thank Mikhail Khovanov and Sergei Gukov for enlight-

ening conversations and exchanges of email.

The first author was supported by the Fundacao para a Ciencia e a Tecnologia

through the programme Programa Operacional Ciencia, Tecnologia, Inovacao

(POCTI), cofinanced by the European Community fund FEDER.


29/29


REFERENCES

[1] D. Bar-Natan, On Khovanovs categorification of the Jones polynomial, Alg. Geom. Top. 2

(2002), 337-370.[2] D. Bar-Natan, Khovanovs homology for tangles and cobordisms, Geom. Topol. 9 (2005), 1443-

1499.

[3] N. M. Dunfield, S. Gukov, J. Rasmussen, The superpolynomial for knot homologies, preprint

2005, math.GT/0505662.

[4] B. Gornik, Note on Khovanov link cohomology, preprint 2004, math.QA/0402266.

[5] S. Gukov and J. Walcher, Matrix factorizations and Kauffman homology, preprint 2005, hep-

th/0512298.

[6] M. Khovanov, A categorification of the Jones polynomial, Duke Math. J. 101(3) (2000), 359-

426.

[7] M. Khovanov, sl(3) link homology, Alg. Geom. Top. 4 (2004), 1045-1081.[8] M. Khovanov, Link homology and Frobenius extensions, preprint 2004, math.QA/0411447

[9] M. Khovanov and L. Rozansky, Matrix factorizations and link homology, preprint 2004,

math.QA/0401268.

[10] G. Kuperberg, Spiders for rank 2 Lie algebras, Comm. Math. Phys. 180(1) (1996), 109-151.[11] E. S. Lee, An endomorphism of the Khovanov invariant, Adv. Math. 197(2) (2005), 554-586.

[12] M. Mackaay, P. Turner, P. Vaz, A remark on Rasmussens invariant of knots, preprint 2005,

math.GT/0509692.

[13] P. Turner, Calculating Bar-Natans characteristic two Khovanov homology, preprint 2004,

math.GT/0411225.

DEPARTAMENTO DE MATEMATICA, UNIVERSIDADE DO ALGARVE, C AMPUS DE GAMBELAS,

8005-139 FARO , P ORTUGAL

E-mail address: [email protected]

DEPARTAMENTO DE MATEMATICA, UNIVERSIDADE DO ALGARVE, C AMPUS DE GAMBELAS,

8005-139 FARO , P ORTUGAL

E-mail address: [email protected]

marco mackaay and pedro vaz- the universal sl3-link homology

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