In this section we give the isomorphism classes of the sl(3)-link homologies, obtained by the
author in joint work with Marco Mackaay in [45]. We just state the main result of Section 3 of [45]. Working over C and taking a, b, c to be complex numbers rather than formal parameters we obtain a filtered theory. Using the same construction as in the first part of this chapter we can define Ua,b,c∗ (L, C), which is the universal sl(3)-homology with coefficients in C (we use one ∗ as superscript to emphasize that it is a singly graded theory).
We could work over Q just as well and obtain the same results, except that in the proofs we would first have to pass to quadratic or cubic field extensions of Q to guarantee the existence of the roots of f (X ) in the field of coefficients of the homology. The arguments presented for
Ua,b,c∗ (L, C) remain valid over those quadratic or cubic extensions. The universal coefficient theorem then shows that our results hold true for the homology defined over Q.
There are three isomorphism classes in Ua,b,c∗ (L, C). The first class is the one to which Khovanov’s original sl(3)-link homology belongs, the second is the one studied by Gornik in the context of matrix factorizations and the last one is new, although in [13] and [22] the
authors make conjectures which are compatible with our results, and can be described in terms of Khovanov’s original sl(2)-link homology.
Theorem 3.2.1. There are three isomorphism classes of Ua,b,c∗ (L, C). For a given choice of a, b, c ∈ C, the isomorphism class of Ua,b,c∗ (L, C) is determined by the number of distinct roots of f(X ) = X3− aX2− bX − c:
1. If f(X ) has one root (with multiplicity three) then Ua,b,c∗ (L, C) is isomorphic to Khovanov’s original sl(3)-link homology, which in our notation is to U0,0,0∗ (L, C).
2. If f(X ) has three distinct roots Ua,b,c∗ (L, C) is isomorphic to Gornik sl(3)-link homology, which in our notation is U0,0,1∗ (L, C).
3. If f(X ) has two distinct roots then
Ua,b,ci (L, C) ∼= M
L0⊆L
HKhi− j(L0),∗(L0, C),
where j(L0) = 2lk(L0, L\L0). This isomorphism does not take into account the internal grading of the Khovanov homology.
We do not know whether in 1 and 2 these isomorphisms preserve the filtration. In 3 the isomorphism does certainly not preserve the filtration, as can be easily seen by computing some
Chapter 4
The foam and the matrix factorization
sl(3) link homologies are equivalent
In this chapter we prove that the universal rational sl(3) link homologies which were constructed in Chapter 3, using foams, and in Section 2.4, using matrix factorizations, are naturally isomor-
phic as projective functors from the category of links and link cobordisms to the category of bigraded vector spaces. For a = b = c = 0 this was conjectured to be true by Khovanov and
Rozansky in [36].
One of the main difficulties one encounters when trying to relate both theories mentioned
above is that the foam approach uses sl(3)-webs (see Section 3.1) whereas the KR theory uses webs (see Section 2.1). In Khovanov and Rozansky’s setup in [36] there is a unique way to
associate a matrix factorization to each web. In general there are several webs that one can associate to an sl(3)-web, so there is no obvious choice of a KR matrix factorization to associate
to an sl(3)-web. However, we show that the KR-matrix factorizations for all webs associated to a fixed sl(3)-web are homotopy equivalent and that between two of them there is a canonical
choice of homotopy equivalence in a certain sense. This allows us to associate an equivalence class of KR-matrix factorizations to each sl(3)-web. After that it is relatively straightforward to
show the equivalence between the foam and the KR sl(3)-link homologies.
In Section 4.1 we change some conventions in the category Foam/`of Chapter 3. Section 4.2 is the core of the chapter. In this section we show how to associate equivalence classes of matrix factorizations to sl(3)-webs and use them to construct a link homology that is equivalent to
Khovanov and Rozansky’s. In Section 4.3 we establish the equivalence between the foam
sl(3)-link homology and the one presented in Section 4.2.
4.1
New normalization in Foam
/`This section contains the modifications in the definition of Foam/`that are necessary to relate it to Khovanov and Rozansky’s universal sl(3) link homology using matrix factorizations.
The modified relations1are defined over Q and are
= a + b + c (3D) = 4 − − − + a + + b (CN) = = 0, = −1 4 (S)
For α, β , δ ≤ 2 we put (see Figure 3.6)
θ (α , β , δ ) = 1 8 (α, β , δ ) = (1, 2, 0) or a cyclic permutation −1 8 (α, β , δ ) = (2, 1, 0) or a cyclic permutation 0 else (Θ)
Using the modified relations `0one can prove the modified identities (RD), (DR). = 2 − (RD) = 2 − (DR) = − − (SqR)
1We thank Scott Morrison for spotting a mistake in the coefficients in a preprint containing the results of this
The Khovanov-Kuperberg relations of Lemma 3.1.9 remain the same with this new norma-
lization.
We now show that the theory obtained with this new normalization is equivalent to the
one in Chapter 3 after tensoring the latter with Q. To distinguish the various constructions let Foam`0 and UQ
a,b,c denote the category Foam/` and the universal homology with the new
normalization. Both are defined over Q[a, b, c]. Let also Modbgdenote the category of bigraded
Q[a, b, c]-modules.
Lemma 4.1.1. The categories Foam`0 andFoam/`⊗
ZQ are isomorphic.
Proof. We define a functor Ξ : Foam`0→ Foam/`⊗ZQ. On objects Ξ is the identity. To define
Ξ on foams we note that the morphisms in both categories are generated by cups, caps, zip, unzips and saddle point cobordisms. We put
7→ 1 2 7→ 1 2 7→ 7→ 1 2 7→ 1 2
It is straightforward to check that Ξ is a well defined isomorphism of categories.
Since Ξ commutes with the differentials of both constructions we have
Corollary 4.1.2. The projective functors UQ
a,b,c and Ua,b,c⊗ZQ from Link to Modbg are na-
turally isomorphic.