Patterns in Khovanov link and chromatic graph homology

TL;DR

This paper explores the relationship between Khovanov link homology and chromatic graph homology, providing improved bounds, explicit formulas, and computational insights into their shared patterns and invariants.

Contribution

It improves bounds on the homological span of chromatic homology and offers explicit formulas for specific homology groups, enhancing understanding of their connection.

Findings

Improved bounds for chromatic homology homological span

Explicit formula for the third chromatic homology group

Computed Khovanov homology and Jones polynomial coefficients for certain links

Abstract

Khovanov homology of a link and chromatic graph homology are known to be isomorphic in a range of homological gradings that depend on the girth of a graph. We discuss patterns shared by these two homology theories. In particular, we improve the bounds for the homological span of chromatic homology by Helme-Guizon, Przytycki and Rong. An explicit formula for the rank of the third chromatic homology group on the main diagonal is given and used to compute the corresponding Khovanov homology group and the fourth coefficient of the Jones polynomial for links with certain diagrams.