TY - JOUR

T1 - General constructions of biquandles and their symmetries

AU - Bardakov, Valeriy

AU - Nasybullov, Timur

AU - Singh, Mahender

N1 - Funding Information: V. Bardakov and T. Nasybullov are supported by the Russian Science Foundation (project 19-41-02005 ). M. Singh is supported by the Swarna Jayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04 . Publisher Copyright: © 2021 Elsevier B.V.

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - Automorphism

KW - Biquandle

KW - Knot invariant

KW - Quandle

KW - Quandle covering

KW - Yang-Baxter equation

UR - http://www.scopus.com/inward/record.url?scp=85118281101&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2021.106936

DO - 10.1016/j.jpaa.2021.106936

M3 - Article

AN - SCOPUS:85118281101

VL - 226

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 7

M1 - 106936

ER -