Quantum Enhancements and Biquandle Brackets

TL;DR

This paper introduces biquandle brackets, a new class of quantum invariants for biquandle colored links, unifying various existing invariants and demonstrating their independence from traditional invariants.

Contribution

The paper presents biquandle brackets, a novel quantum enhancement that unifies skein and cocycle invariants for biquandle colored links, expanding the toolkit for knot invariants.

Findings

Biquandle brackets generalize existing invariants.

They are not determined by traditional invariants like the knot quandle.

Examples show they provide new, independent information about links.

Abstract

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links.Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.