The article is devoted to the construction of a vortex group. This group is a well defined invariant for oriented links in a 3-sphere. It is defined by using generators and relations. The generators are both the crossings of the diagram and two additional formal symbols, while the regions into which the diagram divides the 2-sphere play the role of the relations. It is proved that groups constructed for different diagrams of the same link are isomorphic. A reduced vortex group is obtained from a vortex group by trivialising two specific generators. It is proved that this group allows a balanced presentation. The construction of the reduced vortex group is close to one of the definitions of the Alexander polynomial for links. It is proved that the order of the abelianized reduced vortex group coincides with the determinant of the link.
Keywords: link, vortex group, link determinant
Received May 4, 2025;
Revised June 2, 2025;
Accepted August 13, 2025
Funding Agency: This work was supported by Russian Science Foundation, project № 25-21-20084, https://rscf.ru/project/25-21-20084/.
Philipp Glebovich Korablev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Chelyabinsk State University, Chelyabinsk, 454001 Russia, e-mail: korablev@csu.ru
REFERENCES
1. Bardakov V.G., Mikhalchishina Yu.A., Neshchadim M.V. Virtual link groups. Siberian Math. J., 2017, vol. 58, no. 5, pp. 765–777. https://doi.org/10.1134/S0037446617050032
2. Bardakov V.G., Neshchadim M.V. Knot groups and nilpotent approximability. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, vol. 23, no. 4, pp. 43–51. https://doi.org/10.21538/0134-4889-2017-23-4-43-51
3. Manturov V.O., Nikonov I.M. On braids and groups $G_n^k$. J. Knot Theory and Its Ramific., 2015, vol. 24, no. 13. https://doi.org/10.1142/S0218216515410096
4. Manturov V.O., Fedoseev D.A., Kim S., Nikonov I.M. On groups $G_n^k$ and $\Gamma^k_n$: A study of manifolds, dynamics, and invariants. Bulletin Math. Sci., 2021, vol. 11, no. 02. https://doi.org/10.1142/S1664360721500041
5. Korablev Ph.G. Electric group for knots and links. 2024, 15 p. https://doi.org/10.48550/arXiv.2408.04510
6. Matveev S.V. Distributive groupoids in knot theory. Math. USSR-Sb., 1984, vol. 47, no. 1, pp. 73–83. https://doi.org/10.1070/SM1984v047n01ABEH002630
7. Joyce D. A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra, 1982, vol. 23, pp. 37–65. https://doi.org/10.1016/0022-4049(82)90077-9
8. Polyak M. Minimal generating sets of Reidemeister moves. Quant. Topol., 2010, vol. 1, no. 4, pp. 399–411. https://doi.org/10.4171/QT/10
9. Rolfsen D. Knots and links. Providence, Amer. Math. Soc. Chelsea Publ., 2003, 439 p. https://doi.org/10.1090/chel/346
10. Alexander J.W. Topological invariants of knots and links. Trans. Amer. Math. Soc., 1928, vol. 30, no. 2, pp. 275–306. https://doi.org/10.1090/S0002-9947-1928-1501429-1
11. Crowell R.H., Fox R.H. Introduction to knot theory. NY, 1963, 182 p.
12. Lin X.S., Nelson S. On generalized knot groups. J. Knot Theory and Its Ramific., 2008, vol. 17, no. 3, pp. 263–272. https://doi.org/10.1142/S0218216508006117
13. Wada M. Group invariants of links. Topology, 1992, vol. 31, no. 2, pp. 399–406. https://doi.org/10.1016/0040-9383(92)90029-H
14. Mikhalchishina Yu.A., Generalizations of the Wada representations and virtual link groups. Sib. Math. J., 2017, vol. 58, no. 3, pp. 500–514. https://doi.org/10.1134/S0037446617030132
Cite this article as: Ph.G. Korablev. Vortex group for knots and links in a 3-sphere. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 203–213.
