Chromatic homology, Khovanov homology, and torsion

TL;DR

This paper proves that in certain gradings, Khovanov homology and chromatic homology only contain torsion of order two, and that odd Khovanov homology is torsion-free in its initial gradings, clarifying torsion properties.

Contribution

It establishes that the categorification of the chromatic polynomial only has order two torsion and that odd Khovanov homology is torsion-free in early gradings.

Findings

Chromatic homology only contains torsion of order two.

Khovanov homology only contains torsion of order two in certain gradings.

Odd Khovanov homology is torsion-free in initial gradings.

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 the categorification of the chromatic polynomial only contains torsion of order two, and hence Khovanov homology only contains torsion of order two in the gradings where the isomorphism is defined. We also prove that odd Khovanov homology is torsion-free in its first few homological gradings.