Functoriality of Odd and Generalized Khovanov Homology in $\mathbb{R}^3\times I$

TL;DR

This paper extends the generalized Khovanov bracket to smooth link cobordisms in 3-space cross an interval, establishing functoriality up to scalars, and explores the functorial properties of odd Khovanov homology.

Contribution

It introduces a functorial extension of the generalized Khovanov bracket to cobordisms in $ ext{R}^3 imes I$, clarifying functoriality for odd and even variants.

Findings

Generalized Khovanov bracket is functorial up to scalars in $ ext{R}^3 imes I$.

Odd Khovanov homology is functorial up to sign.

Odd Khovanov homology is not functorial in $S^3 imes I$.

Abstract

We extend the generalized Khovanov bracket to smooth link cobordisms in $\mathbb{R}^3\times I$ and prove that the resulting theory is functorial up to global invertible scalars. The generalized Khovanov bracket can be specialized to both even and odd Khovanov homology. Particularly by setting $\pi=-1$, we obtain that odd Khovanov homology is functorial up to sign. We end by showing that odd Khovanov homology is not functorial under smooth link cobordisms in $S^3\times I$.