The Functoriality of Odd Khovanov Homology up to Sign and Applications

TL;DR

This paper extends odd Khovanov homology to link cobordisms, establishes its functoriality up to sign, and explores algebraic structures and applications, including actions of the Hecke algebra on cable knots.

Contribution

It introduces a functorial odd Khovanov theory for dotted link cobordisms and demonstrates its applications to algebraic actions on knot homologies.

Findings

Functorial extension of odd Khovanov homology up to sign.

Construction of a module structure over the exterior algebra.

Action of the Hecke algebra on the homology of cable knots.

Abstract

In this dissertation, we extend the odd Khovanov bracket to link cobordisms and prove that our construction is functorial up to sign. We then build an odd Khovanov theory for dotted link cobordisms. Out of the dotted theory, a module structure on the odd Khovanov homology of a diagram over the exterior algebra of the diagram's coloring group arises. We finish by using our functoriality result to prove that if $n$ is even or if the knot has even framing, then the odd Khovanov homology of the $n$-cable of a knot admits an action of the Hecke algebra $\mathcal{H}(q^2,n)$ at $q=i$.