Bikei, Involutory Biracks and unoriented link invariants

TL;DR

This paper introduces involutory biracks, a new algebraic structure that defines invariants for unoriented links, including classical and virtual links, expanding the tools for knot theory analysis.

Contribution

The paper defines involutory biracks and bikei, establishing their role in constructing counting invariants for unoriented links, including virtual knots.

Findings

Involutory biracks generalize biquandles for unoriented links.

Counting invariants can detect non-invertibility of virtual knots.

Example of a non-involutory birack distinguishing virtual knot properties.

Abstract

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which we call bikei. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.