A volume form on the Khovanov invariant

TL;DR

This paper introduces a volume form on Khovanov homology derived from Reidemeister torsion, providing a new invariant that captures algebraic torsion information and remains invariant under Reidemeister moves.

Contribution

It constructs and proves the invariance of a volume form on Khovanov homology using Reidemeister torsion, offering a novel numerical invariant for knots and links.

Findings

The volume form is invariant under Reidemeister moves.

Examples demonstrate the volume form captures algebraic torsion.

Provides a new numerical invariant for knots and links.

Abstract

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over $\mathbb{Z}$ is recovered.