General constructions of biquandles and their symmetries

General constructions of biquandles and their symmetries Full article

Journal

Journal of Pure and Applied Algebra
ISSN: 0022-4049

Output data

Year: 2022, Volume: 226, Number: 7, Article number : 106936, Pages count : DOI: 10.1016/j.jpaa.2021.106936

Tags

Automorphism; Biquandle; Knot invariant; Quandle; Quandle covering; Yang-Baxter equation

Authors

Bardakov V. ^1,2,3,4 , Nasybullov T. ^1,2,3 , Singh M. ⁵

Affiliations

1	Tomsk State University, pr. Lenina 36, Tomsk, 634050, Russian Federation
2	Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, Novosibirsk, 630090, Russian Federation
3	Novosibirsk State University, Pirogova 1, Novosibirsk, 630090, Russian Federation
4	Novosibirsk State Agricultural University, Dobrolyubova 160, Novosibirsk, 630039, Russian Federation
5	Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81 Punjab, S. A. S. Nagar, P. O. Manauli, 140306, India

Funding (1)

Russian Science Foundation

19-41-02005

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.

Bardakov V. , Nasybullov T. , Singh M.
General constructions of biquandles and their symmetries
Journal of Pure and Applied Algebra. 2022. V.226. N7. 106936 . DOI: 10.1016/j.jpaa.2021.106936 WOS Scopus РИНЦ OpenAlex

Submitted:	Jun 18, 2021
Published online:	Oct 21, 2021
Published print:	Jul 5, 2022

Web of science:	WOS:000780273400005
Scopus:	2-s2.0-85118281101
Elibrary:	47518308
OpenAlex:	W3205631036

DB	Citing
Scopus	5
Web of science	5
OpenAlex	8
Elibrary	2