Chromatic homology, Khovanov homology, and torsion

Abstract

In the first few homological gradings, there is an isomorphism between the Khovanov homology of a link and the categorification of the chromatic polynomial of a graph related to the link. In this article, we show that all torsion in the categorification of the chromatic polynomial is of order two, and hence all torsion in Khovanov homology in the gradings where the isomorphism is defined is of order two. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.

Patterns in Khovanov link and chromatic graph homology

“…The minimal i-grading with torsion is either i = 1 (odd cycle) or i = 2 (bipartite) [PPS09]. On the other hand, H A2 (G) contains one Z 2 in (i + 1, j − 1) for each (i, j), (i + 1, j − 2) knight move pair, based on the proof of Theorem 2 [LS17]. Therefore, the maximal homological grading with torsion is…”

“…Similarities between Khovanov link and chromatic homology go beyond this theorem, and extend mainly to alternating knots and their associated graphs. Note that the following result from [LS17] states that the portion of Khovanov homology of any link is the same as Khovanov homology of an alternating link provided that their associated graphs are isomorphic. More precisely, if D is an alternating diagram of a link L and D ′ is a diagram of any link L such that G = G + (D) = G + (D ′ ), then we have the following isomorphism of Khovanov homology groups:…”

“…As a corollary, we will be able to partially describe Khovanov homology of alternating 3-strand pretzel knots. For thin pretzel links, such as those which are alternating or quasi-alternating ([OS08], [Gre10]), torsion is determined by the Jones polynomial and signature via results of Alex Shumakovitch that inspired results in [LS17]. Three-strand pretzel links of the form (p 1 , p 2 , −q) are quasi-alternating if and only if q > min{p 1 , p 2 } [Gre10].…”

“…We use recent results from [LS17] and the formulas for coefficients of P G (λ) given in [Far80], to calculate the torsion in (4, v − 4) grading as well. Note that [Far80, Theorem 2] can be used to compute torsion in degree (5, v − 5) of chromatic homology.…”

“…There is a partial isomorphism between Khovanov homology of a semi-adequate link L and the chromatic homology of a state graph G + (D) obtained from a diagram D of L. The extent of the isomorphism depends only on the length of the shortest cycle in G + (D). Chromatic homology over the algebra A 2 = Z[x]/(x 2 ) has only Z 2 torsion, and is equivalent to the chromatic polynomial [LS17]. When other polynomial algebras of the form A m = Z[x]/(x m ) are used in the construction, the resulting homologies may be stronger than the chromatic polynomial and may contain torsion of arbitrary order [PPS09].…”