The Bar-Natan Theory Splits

TL;DR

This paper proves that over the binary field, the Bar-Natan perturbation of Khovanov homology decomposes into two isomorphic reduced theories, extending known results for unperturbed Khovanov homology.

Contribution

It demonstrates that the Bar-Natan perturbation splits into two reduced, isomorphic theories over _2, generalizing Shumakovitch's result to the perturbed case.

Findings

The Bar-Natan perturbation splits as a direct sum over _2.

The two reduced theories are isomorphic.

Extension of Shumakovitch's result to the perturbed setting.

Abstract

We show that over the binary field $\mathbb F_2$, the Bar-Natan perturbation of Khovanov homology splits as the direct sum of its two reduced theories, which we also prove are isomorphic. This extends Shumakovitch's analogous result for ordinary Khovanov homology, without the perturbation.