The universalsl₃–link homology

Abstract

We define the universal sl 3 -link homology, which depends on 3 parameters, following Khovanov's 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 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. Following an approach similar to Gornik's we show that this new link homology can be described in terms of Khovanov's original sl 2 -link homology.57M27; 57M25, 81R50, 18G60

Functoriality for the su3Khovanov homology

“…The categorified su 3 invariant was shown to be functorial up to a sign by Mackaay and Vaz [11] using an argument of Bar-Natan [1]. However, the representation theoretic problem for su 2 does not exist for su 3 , whose fundamental representations are dual to each other, but not self-dual.…”

Some differentials on Khovanov–Rozansky homology

“…(This idea has its source in the work of Gornik [4]. Although we will not pursue it here, some supporting evidence has been provided by Mackaay and Vaz [16].) In particular, it seems unlikely that the set of all homologies H p (L, i), p ∈ Q[x] contains more information than is present in the sl(N ) homologies.…”

“…The knots 9 16 , 9 22 , 9 24 , 9 25 , 9 28 , 9 29 , 9 30 , and 9 32 -9 49 are not two-bridge. Only one -9 16 -is the closure of a threestrand braid, and Webster's program shows that it is KR-thin. Of the rest, all but five -9 29 ,9 42 ,9 43 , 9 46 , and 9 47 -can be shown to be KR-thin using the criterion of Corollary 7.7.…”

sl(2) tangle homology with a parameter and singular cobordisms

“…Instead of .1 C 1/-dimensional cobordisms he uses webs and foams modulo a finite set of relations. Following his approach, Mackaay and Vaz [14] defined the universal sl.3/ link homology which depends on 3 parameters, and their theory arises from a Frobenius system corresponding to ZOEX; a; b; c=.X 3 aX 2 bX c/:…”

“…The construction follows closely the work by Bar-Natan [1], with the main difference that we keep track of orientations. More precisely, we start from the oriented state model for the Jones polynomial and work with webs and foams modulo local relations, much as it is done in [10] and [14], but instead of using planar trivalent graphs we consider bivalent graphs. Consequently, our foams are 2-dimensional CW-complexes with singularities where two 2-cells are joining, as opposed to three 2-cells.…”