Biquandle Brackets and Knotoids

TL;DR

This paper introduces biquandle brackets as a quantum enhancement of biquandle invariants for knots and links, demonstrating their effectiveness through examples that show they outperform previous invariants.

Contribution

The paper extends biquandle invariants by incorporating biquandle brackets, providing a new, stronger invariant for oriented knots and links.

Findings

Biquandle brackets enhance the biquandle counting invariant.

Examples show the new invariants are stronger than previous ones.

Methodology for calculating biquandle brackets is illustrated.

Abstract

Biquandle brackets are a type of quantum enhancement of the biquandle counting invariant for oriented knots and links, defined by a set of skein relations with coefficients which are functions of biquandle colors at a crossing. In this paper we use biquandle brackets to enhance the biquandle counting matrix invariant defined by the first two authors in arXiv:1803.11308. We provide examples to illustrate the method of calcuation and to show that the new invariants are stronger than the previous ones.