Categorifying Biquandle Brackets

TL;DR

This paper explores a Khovanov homology-inspired approach to categorify biquandle brackets, resulting in a new knot invariant that generalizes Khovanov homology but does not fully categorify biquandle brackets.

Contribution

It proposes a novel homology-style construction for biquandle brackets, introducing a canonical biquandle 2-cocycle linked to these brackets.

Findings

The construction generalizes Khovanov homology.

It does not always recover the original biquandle bracket.

A new canonical biquandle 2-cocycle is defined.

Abstract

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones polynomial, which is categorified by Khovanov homology. At the end of their paper, Nelson, Orrison, and Rivera asked if the methods of Khovanov homology could be extended to obtain a categorification of biquandle brackets. We outline herein a Khovanov homology-style construction that is an attempt to obtain such a categorification of biquandle brackets. The resulting knot invariant generalizes Khovanov homology, but the biquandle bracket is not always recoverable, meaning the construction is not a true categorification of biquandle brackets. However, the construction does lead to a definition that gives a "canonical" biquandle 2-cocycle associated to a biquandle bracket, which, to the authors' knowledge, was not previously known.