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Bridging colorings of virtual links from virtual biquandles to biquandles

Published online by Cambridge University Press:  14 March 2025

Mohamed Elhamdadi  and
Manpreet Singh Open the ORCID record for Manpreet Singh [Opens in a new window]
Mohamed Elhamdadi
Affiliation:
Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue, Tampa, FL, 33620, United States e-mail: [email protected]
Manpreet Singh*
Affiliation:
Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Avenue, Tampa, FL, 33620, United States e-mail: [email protected]
*
e-mail: [email protected] [email protected]
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Abstract

A biquandle is a solution to the set-theoretical Yang–Baxter equation, which yields invariants for virtual knots such as the coloring number and the state-sum invariant. A virtual biquandle enriches the structure of a biquandle by incorporating an invertible unary map. This unary operator plays a crucial role in defining the action of virtual crossings on the labels of incoming arcs in a virtual link diagram. This leads to extensions of invariants from biquandles to virtual biquandles, thereby enhancing their strength.

In this article, we establish a connection between the coloring invariant derived from biquandles and virtual biquandles. We prove that the number of colorings of a virtual link L by virtual biquandles can be recovered from colorings by biquandles. We achieve this by proving the equivalence between two different representations of virtual braid groups. Furthermore, we introduce a new set of labeling rules using which one can construct a presentation of the associated fundamental virtual biquandle of L using only the relations coming from the classical crossings. This is an improvement to the traditional method, where writing down a presentation of the associated fundamental virtual biquandle necessitates noting down the relations arising from the classical and virtual crossings.

Keywords

Virtual knotsvirtual braidsbiquandlesvirtual biquandles
Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 14
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

M.E. is partially supported by Simons Foundation collaboration grant 712462. M.S. has received funding from Fulbright-Nehru Postdoctoral Fellowship grant 2865/FNPDR/2022.

References

Bardakov, V. G. and Bellingeri, P., Combinatorial properties of virtual braids . Topol. Appl. 156(2009), no. 6, 1071–1082.Google Scholar
Bardakov, V. G., Mikhalchishina, Y. A., and Neshchadim, M. V., Representations of virtual braids by automorphisms and virtual knot groups . J. Knot Theory Ramif. 26(2017), no. 1, 1750003, 17.CrossRefGoogle Scholar
Bardakov, V. G., Neshchadim, M. V., and Singh, M., Virtually symmetric representations and marked Gauss diagrams . Topol. Appl. 306(2022), Paper No. 107936, 24.CrossRefGoogle Scholar
Boden, H. U. and Chrisman, M., Virtual concordance and the generalized Alexander polynomial . J. Knot Theory Ramif. 30(2021), no. 5, Paper No. 2150030, 35.CrossRefGoogle Scholar
Boden, H. U., Gaudreau, R., Harper, E., Nicas, A. J., and White, L., Virtual knot groups and almost classical knots . Fund. Math. 238(2017), no. 2, 101–142.CrossRefGoogle Scholar
Boden, H. U. and Nagel, M., Concordance group of virtual knots . Proc. Amer. Math. Soc. 145(2017), no. 12, 5451–5461.CrossRefGoogle Scholar
Carter, J. S., Kamada, S., and Saito, M., Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong). J. Knot Theory Ramifications 11(2002), no. 3, pp. 311–322.Google Scholar
Ceniceros, J. and Nelson, S., Virtual Yang-Baxter cocycle invariants . Trans. Amer. Math. Soc. 361(10):5263–5283, 2009.Google Scholar
Dye, H. A. and Kauffman, L. H., Minimal surface representations of virtual knots and links . Algebr. Geom. Topol. 5(2005), 509–535.CrossRefGoogle Scholar
Elhamdadi, M. and Singh, M., Twisted Yang-Baxter sets, cohomology theory, and application to knots. Preprint, 2024, arXiv:2401.01533.Google Scholar
Farinati, M. A. and García Galofre, J., Virtual link and knot invariants from non-abelian Yang-Baxter 2-cocycle pairs . Osaka J. Math. 56(2019), no. 3, 525–547.Google Scholar
Fenn, R., Jordan-Santana, M., and Kauffman, L., Biquandles and virtual links . Topology Appl. 145(2004), nos. 1–3, 157–175.CrossRefGoogle Scholar
Fenn, R. A. and Bartholomew, A., Erratum: Biquandles of small size and some invariants of virtual and welded knots . J. Knot Theory Ramif. 26(2017), no. 8, 1792002, 11.Google Scholar
Hrencecin, D. and Kauffman, L. H., Biquandles for virtual knots . J. Knot Theory Ramif. 16(2007), no. 10, 1361–1382.CrossRefGoogle Scholar
Jaeger, F., Kauffman, L. H., and Saleur, H., The Conway polynomial in R3 and in thickened surfaces: a new determinant formulation . J. Combin. Theory Ser. B 61(1994), no. 2, 237–259.CrossRefGoogle Scholar
Kamada, N., On the Jones polynomials of checkerboard colorable virtual links . Osaka J. Math. 39(2002), no. 2, 325–333.Google Scholar
Kamada, N., Span of the Jones polynomial of an alternating virtual link . Algebr. Geom. Topol. 4(2004), 1083–1101.Google Scholar
Kamada, N. and Kamada, S., Abstract link diagrams and virtual knots . J. Knot Theory Ramif. 9(2000), no. 1, 93–106.CrossRefGoogle Scholar
Kamada, S., Braid presentation of virtual knots and welded knots . Osaka J. Math. 44(2007), no. 2, 441–458.Google Scholar
Kauffman, L. and Lambropoulou, S., The $L$ -move and virtual braids. In: Intelligence of low dimensional topology 2006, J. S. Carter, S. Kamada, L. H. Kauffman, A. Kawauchi, and T. Kohno (eds), Series on Knots and Everything, 40. World Scientific Publishing, Hackensack, NJ, 2007, pp. 133–142.CrossRefGoogle Scholar
Kauffman, L. H., Virtual knot theory . Eur. J. Combin. 20(1999), no. 7, 663–690.CrossRefGoogle Scholar
Kauffman, L. H. and Lambropoulou, S., Virtual braids . Fund. Math. 184(2004), 159–186.Google Scholar
Kauffman, L. H. and Manturov, V. O., Virtual biquandles . Fund. Math. 188(2005), 103–146.Google Scholar
Kuperberg, G., What is a virtual link? Algebr. Geom. Topol. 3(2003), 587–591.CrossRefGoogle Scholar
Manturov, V. O., On invariants of virtual links . Acta Appl. Math. 72(2002), no. 3, 295–309.CrossRefGoogle Scholar
Silver, D. S. and Williams, S. G., Alexander groups and virtual links . J. Knot Theory Ramif. 10(2001), no. 1, 151–160.CrossRefGoogle Scholar
Wada, M., Group invariants of links . Topology 31(1992), no. 2, 399–406.Google Scholar