The topological biquandle of a link

TL;DR

This paper introduces the topological biquandle, a new algebraic structure associated with an oriented link, providing a topological perspective and relating it to the fundamental biquandle.

Contribution

It defines the topological biquandle for links, presents its algebraic structure, and explains its relation to the fundamental biquandle, extending link invariants.

Findings

Defined the topological biquandle for oriented links

Provided a presentation of the topological biquandle

Established the relation to the fundamental biquandle

Abstract

To every oriented link $L$, we associate a topologically defined biquandle $\widehat{\mathcal{B}}_{L}$, which we call the topological biquandle of $L$. The construction of $\widehat{\mathcal{B}}_{L}$ is similar to the topological description of the fundamental quandle given by Matveev. We find a presentation of the topological biquandle and explain how it is related to the fundamental biquandle of the link.