General constructions of biquandles and their symmetries

TL;DR

This paper introduces new methods for constructing biquandles from groups and quandles, explores their properties, and analyzes their automorphism groups, providing solutions to the Yang-Baxter equation relevant to virtual knot theory.

Contribution

It presents novel constructions of biquandles from groups and quandles, including the holomorph biquandle, and characterizes their automorphism groups, advancing the algebraic understanding of biquandles.

Findings

Complete classification of words in free groups yielding biquandles

Construction of biquandles on unions and products of quandles

Automorphism groups of biquandles related to underlying quandles

Abstract

Biquandles are algebraic objects with two binary operations whose axioms encode the generalized Reidemeister moves for virtual knots and links. These objects also provide set-theoretic solutions of the well-known Yang-Baxter equation. The first half of this paper proposes some natural constructions of biquandles from groups and from their simpler counterparts, namely, quandles. We completely determine all words in the free group on two generators that give rise to (bi)quandle structures on all groups. We give some novel constructions of biquandles on unions and products of quandles, including what we refer as the holomorph biquandle of a quandle. These constructions give a wealth of solutions of the Yang-Baxter equation. We also show that for nice quandle coverings a biquandle structure on the base can be lifted to a biquandle structure on the covering. In the second half of the paper, we determine automorphism groups of these biquandles in terms of associated quandles showing elegant relationships between the symmetries of the underlying structures.