Research ArticleNo Access

The geometric realization of a normalized set-theoretic Yang–Baxter homology of biquandles

Xiao Wang

Department of Mathematics, Jilin University, Changchun 130012, China

E-mail Address: wangxiaotop@jlu.edu.cn

Search for more papers by this author

and

Seung Yeop Yang

Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea

E-mail Address: seungyeop.yang@knu.ac.kr

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0218216522500511Cited by:0 (Source: Crossref)

Abstract

Biracks and biquandles, which are useful for studying the knot theory, are special families of solutions of the set-theoretic Yang–Baxter equation. A homology theory for the set-theoretic Yang–Baxter equation was developed by Carter et al. in order to construct knot invariants. In this paper, we construct a normalized (co)homology theory of a set-theoretic solution of the Yang–Baxter equation. We obtain some concrete examples of nontrivial $n$ $n$ -cocycles for Alexander biquandles. For a biquandle $X,$ $X,$ its geometric realization $B X$ $B X$ is discussed, which has the potential to build invariants of links and knotted surfaces. In particular, we demonstrate that the second homotopy group of $B X$ $B X$ is finitely generated if the biquandle $X$ $X$ is finite.

Keywords:

AMSC: 55N35, 55Q52, 57K10

References

1. R. J. Baxter , Partition function of the eight-vertex lattice model, Ann. Phys. 70 (1972) 193–228. Crossref, Web of Science, Google Scholar
2. J. S. Carter, M. Elhamdadi and M. Saito , Homology theory for the set-theoretic Yang-Baxter equation and knot invariants from generalizations of quandles, Fund. Math. 184 (2004) 31–54. Crossref, Web of Science, Google Scholar
3. J. Ceniceros, M. Elhamdadi, M. Green and S. Nelson , Augmented biracks and their homology, Int. J. Math. 25(9) (2014) 1450087. Link, Web of Science, Google Scholar
4. J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito , Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc. 355(10) (2003) 3947–3989. Crossref, Web of Science, Google Scholar
5. A. S. Crans, S. Mukherjee and J. H. Przytycki , On homology of associative shelves, J. Homotopy Relat. Struct. 12(3) (2017) 741–763. Crossref, Web of Science, Google Scholar
6. V. G. Drinfel’d , On some unsolved problems in quantum group theory, Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, Vol. 1510 (Springer, Berlin, 1992), pp. 1–8. Crossref, Google Scholar
7. P. Etingof and S. Gelaki , A method of construction of finite-dimensional triangular semisimple Hopf algebras, Math. Res. Lett. 5(4) (1998) 551–561. Crossref, Web of Science, Google Scholar
8. P. Etingof, T. Schedler and A. Soloviev , Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100(2) (1999) 169–209. Crossref, Web of Science, Google Scholar
9. S. Eilenberg and J. A. Zilber , Semi-simplicial complexes and singular homology, Ann. Math. 51 (1950) 499–513. Crossref, Web of Science, Google Scholar
10. R. Fenn , Tackling the trefoils, J. Knot Theory Ramifications 21(13) (2012) 1240004. Link, Web of Science, Google Scholar
11. R. Fenn, C. Rourke and B. Sanderson , An introduction to species and the rack space, in Topics in Knot Theory, NATO Science Series C: Mathematical and Physical Sciences, Vol. 399 (Kluwer Academic Publisher, Dordrecht, 1993) pp. 33–55. Crossref, Google Scholar
12. R. Fenn, C. Rourke and B. Sanderson , Trunks and classifying spaces, Appl. Categ. Struct. 3(4) (1995) 321–356. Crossref, Web of Science, Google Scholar
13. K. Ishikawa and K. Tanaka, Quandle colorings vs. biquandle colorings, preprint (2012) arXiv:1912.12917. Google Scholar
14. V. F. R. Jones , Hecke algebra representations of braid groups and link polynomials, Ann. Math. 126(2) (1987) 335–388. Crossref, Web of Science, Google Scholar
15. D. Joyce , A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra 23(1) (1982) 37–65. Crossref, Web of Science, Google Scholar
16. V. Lebed and L. Vendramin , Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math. 304 (2017) 1219–1261. Crossref, Web of Science, Google Scholar
17. J.-H. Lu, M. Yan, and Y. Zhu , On Hopf algebras with positive bases, J. Algebra 237(2) (2001) 421–445. Crossref, Web of Science, Google Scholar
18. J.-H. Lu, M. Yan, and Y. Zhu , On the set-theoretical Yang-Baxter equation, Duke Math. J. 104(1) (2000) 1–18. Crossref, Web of Science, Google Scholar
19. J.-L. Loday , Cyclic Homology, 2nd edn., Grundlehren der mathematischen Wissenschaften, Vol. 301 (Springer, 1998). Crossref, Google Scholar
20. S. V. Matveev, Distributive groupoids in knot theory, Mat. Sb. (N.S.) 119(1) (1982) 78–88. English translation: Math. USSR-Sb. 47(1) (1984) 73–83. Google Scholar
21. J. P. May , Simplicial Objects in Algebraic Topology, Chicago Lectures in Mathematics (University of Chicago Press, 1967). Google Scholar
22. J. McCleary , A User’s Guide to Spectral Sequences, Cambridge Studies in Advanced Mathematics, Vol. 58 (Cambridge University Press, Cambridge, 2001). Google Scholar
23. T. Nosaka , On homotopy groups of quandle spaces and the quandle homotopy invariant of links, Topol. Appl. 158(8) (2011) 996–1011. Crossref, Web of Science, Google Scholar
24. T. Nosaka , Quandle homotopy invariants of knotted surfaces, Math. Z. 274(1–2) (2013) 341–365. Google Scholar
25. J. H. Przytycki , Knots and distributive homology: From arc colorings to Yang–Baxter homology, in New Ideas in Low Dimensional Topology, Series Knots Everything, Vol. 56 (World Scientific, Hackensack, New Jersey 2015), pp. 413–488. Link, Google Scholar
26. J. H. Przytycki, P. Vojtěchovský, and S. Y. Yang, Set-theoretic Yang–Baxter (co)homology theory of involutive non-degenerate solutions, preprint (2003), arXiv:1911.03009. Google Scholar
27. J. H. Przytycki and X. Wang , Equivalence of two definitions of set-theoretic Yang-Baxter homology and general Yang–Baxter homology, J. Knot Theory Ramifications 27(7) (2018) 1841013. Link, Web of Science, Google Scholar
28. J.-P. Serre, Homologie singulière des espaces fibrés. Applications, PhD thesis Université Paris IV-Sorbonne (1951). Google Scholar
29. V. G. Turaev , The Yang–Baxter equation and invariants of links, Invent. Math. 92(3) (1988) 527–553. Crossref, Web of Science, Google Scholar
30. A. Weinstein and P. Xu , Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys. 148(2) (1992) 309–343. Crossref, Web of Science, Google Scholar
31. C. N. Yang , Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967) 1312–1315. Crossref, Web of Science, Google Scholar
32. S. Y. Yang , Extended quandle spaces and shadow homotopy invariants of classical links, J. Knot Theory Ramifications 26(3) (2017) 1741010. Link, Web of Science, Google Scholar