A sign assignment in totally twisted Khovanov homology

TL;DR

This paper extends totally twisted Khovanov homology from characteristic 2 to integer coefficients, providing a new complex that computes reduced odd Khovanov homology with an explicit differential up to a sign ambiguity.

Contribution

It introduces an integer coefficient version of totally twisted Khovanov homology and relates it to a spanning-tree complex with an explicit differential.

Findings

Constructed a complex with integer coefficients for twisted Khovanov homology.

Established the equivalence to a spanning-tree complex.

Provided an explicit differential modulo a sign ambiguity.

Abstract

We lift the characteristic-2 totally twisted Khovanov homology of Roberts and Jaeger to a theory with integer coefficients. The result is a complex computing reduced odd Khovanov homology for knots. This complex is equivalent to a spanning-tree complex whose differential is explicit modulo a sign ambiguity coming from the need to choose a sign assignment in the definition of odd Khovanov homology.