Torsion in Khovanov homology of semi-adequate links

TL;DR

This paper investigates the presence of Z_2-torsion in the Khovanov homology of semi-adequate links, providing explicit formulas and linking torsion to graph properties, thus advancing understanding of link invariants.

Contribution

It introduces explicit formulas for torsion in Khovanov homology of semi-adequate links and connects torsion existence to graph cycles and braid index.

Findings

Z_2-torsion exists if the Tait graph has a cycle of length at least 3

Torsion of odd order also exists, but lacks a general theoretical explanation

Torsion presence may be related to braid index

Abstract

The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which provides a link between the link homology and well-developed theory of Hochschild homology. In particular, we obtain explicit formulae for torsion and prove that Khovanov homology of semi-adequate links contains $Z_2$-torsion if the corresponding Tait-type graph has a cycle of length at least 3. Computations show that torsion of odd order exists but there is no general theory to support these observations. We conjecture that the existence of torsion is related to the braid index.