# The Structure of Biquandle Brackets

## TL;DR

This paper explores the algebraic structure of biquandle brackets, revealing their connection to biquandle 2-cocycles and demonstrating that certain invariants are equivalent to the Jones polynomial for knots.

## Contribution

It proves that biquandle brackets multiplied by a function are related to biquandle 2-cocycles and shows the equivalence of a new invariant to the Jones polynomial for knots.

## Key findings

- A biquandle bracket multiplied by a function is a biquandle 2-cocycle.
- A new invariant factors through biquandle 2-cocycles and equals the Jones polynomial on knots.
- Provides new insights into the structure of biquandle brackets and their algebraic properties.

## 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. We prove herein that if a biquandle bracket is the pointwise product of another biquandle bracket with some function $\phi$, then $\phi$ is a a biquandle 2-cocycle (up to a constant multiple). As an application, we show that a new invariant introduced by Yang factors in this way, which allows us to show that the new invariant is in fact equivalent to the Jones polynomial on knots. Additionally, we provide a few new results about the structure of biquandle brackets and their relationship with biquandle 2-cocycles.