In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for $S^{1}$ -oriented handlebody-links using $2$ -cocycles. When a multiple conjugation biquandle $X \times ℤ_{type X_{Y}}$ is obtained from a biquandle $X$ using $n$ -parallel operations, we provide a $2$ -cocycle (or $3$ -cocycle) of the multiple conjugation biquandle $X \times ℤ_{type X_{Y}}$ from a $2$ -cocycle (or $3$ -cocycle) of the biquandle $X$ equipped with an $X$ -set $Y$ .

Keywords:

AMSC: 57M27, 57M25

References

1. 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, ISI, Google Scholar
2. S. Carter, A. Ishii, M. Saito and K. Tanaka, Homology for quandles with partial group operations, Pacific J. Math. 287(1) (2017) 19–48. Crossref, Google Scholar
3. 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 (2003) 3947–3989. Crossref, ISI, Google Scholar
4. R. Fenn, C. Rourke and B. Sanderson, Trunks and classifying spaces, Appl. Categ. Structures 3 (1995) 321–356. Crossref, ISI, Google Scholar
5. A. Ishii, Moves and invariants for knotted handlebodies, Algebr. Geom. Topol. 8 (2008) 1403–1418. Crossref, ISI, Google Scholar
6. A. Ishii, A multiple conjugation quandle and handlebody-knots, Topology Appl. 196 (2015) 492–500. Crossref, Google Scholar
7. A. Ishii, The Markov theorem for spatial graphs and handlebody-knots with Y-orientations, Internat. J. Math. 26 (2015), Article ID: 1550116, 23 pp. Link, Google Scholar
8. A. Ishii and M. Iwakiri, Quandle cocycle invariants for spatial graphs and knotted handlebodies, Canad. J. Math. 64 (2012) 102–122. Crossref, ISI, Google Scholar
9. A. Ishii, M. Iwakiri, Y. Jang and K. Oshiro, A $G$ -family of quandles and handlebody-knots, Illinois J. Math. 57 (2013) 817–838. Crossref, Google Scholar
10. A. Ishii, M. Iwakiri, S. Kamada, J. Kim, S. Matsuzaki and K. Oshiro, A multiple conjugation biquandles and handlebody-links, Hiroshima Math. J. 48 (2018) 89–117. Crossref, Google Scholar
11. A. Ishii and S. Nelson, Partially multiplicative biquandles and handlebody-knots, Contemp. Math. 689 (2017) 159–176. Crossref, Google Scholar
12. L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras, and generalized Alexander polynomial for virtual links, Contemp. Math. 318 (2003) 113–140. Crossref, Google Scholar
13. 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, ISI, Google Scholar
14. S. Nelson and K. Pelland, Birack shadow modules and their link invariants, J. Knot Theory Ramifications 22(10) (2013), Article ID: 1350056, 12 pp. Link, ISI, Google Scholar
15. T. Nosaka, Quandle cocycles from invariant theory, Adv. Math. 245 (2013) 423–438. Crossref, ISI, Google Scholar

Published: 3 September 2018

- Cited By 1
  - Cocycles of G-Alexander biquandles and G-Alexander multiple conjugation biquandles
    Atsushi Ishii, Masahide Iwakiri, Seiichi Kamada, Jieon Kim and Shosaku Matsuzaki et al.
    1 Dec 2020 | Topology and its Applications
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Vol. 27, No. 11
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History

Received 20 January 2018

Revised 23 April 2018

Accepted 24 April 2018
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Biquandle (co)homology and handlebody-links

Abstract

References

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