General constructions of biquandles and their symmetries |
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| Institution: | 1. Tomsk State University, pr. Lenina 36, 634050 Tomsk, Russia;2. Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia;3. Novosibirsk State University, Pirogova 1, 630090 Novosibirsk, Russia;4. Novosibirsk State Agricultural University, Dobrolyubova 160, 630039 Novosibirsk, Russia;5. Department of Mathematical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81, S. A. S. Nagar, P. O. Manauli, 140306, Punjab, India;1. Department of Mathematics, University of Minnesota, Twin Cities, Vincent Hall, 206 Church St SE, Minneapolis, MN 55455, United States of America;2. Department of Mathematics and Statistics, Colby College, Davis Building, Second Floor, 5830 Mayflower Hill, Waterville, ME 04901, United States of America;1. Technische Universität Dortmund, Fakultät für Mathematik, Lehrstuhl LSVI, Vogelpothsweg 87, 44227 Dortmund, Germany;2. NTU-MATH, Astronomy Mathematics Building 5F, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan, ROC;1. Institute for Theoretical Sciences, Westlake Institute for Advanced Study, Westlake University, Hangzhou, Zhejiang, 310024, China;2. Department of Mathematics, The University of Uath, Salt Lake City, UT 84112, USA;1. School of Mathematical Sciences, University of the Chinese Academy of Sciences, Beijing 100049, China;2. College of Mathematics and Statistics, Shenzhen University, Shenzhen, 518060, Guangdong, China |
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| 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. |
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| Keywords: | Quandle Biquandle Quandle covering Automorphism Knot invariant Yang-Baxter equation |
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