Odd Khovanov homology

TL;DR

This paper introduces a new link invariant related to Khovanov homology, replacing symmetric algebra with exterior algebra, providing a refined invariant that differs over rationals but matches mod 2.

Contribution

It presents a novel construction of a link invariant using exterior algebra, expanding the framework of Khovanov homology with new algebraic structures.

Findings

The invariant matches Khovanov homology mod 2.

It differs from Khovanov homology over the rationals.

The reduced version's Euler characteristic equals the normalized Jones polynomial.

Abstract

We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two invariants have the same reduction modulo 2, but differ over the rationals. There is a reduced version which is a link invariant whose graded Euler characteristic is the normalized Jones polynomial.