Abstract
A multiple conjugation quandle is an algebraic structure whose axioms are motivated from handlebody-knot theory. By using an MCQ Alexander pair f, which is a pair of maps corresponding to a linear extension of a multiple conjugation quandle, we can construct the f-twisted Alexander matrices, which produce invariants for handlebody-knots. The purpose of this paper is to show the sufficiency of the definition of MCQ Alexander pairs in constructing the f-twisted Alexander matrices from the aspect of linear extensions of multiple conjugation quandles. Furthermore, we introduce the notion of cohomologous for MCQ Alexander pairs, which induces the same invariant for handlebody-knots.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Change history
22 August 2022
The original article is revised to update the corrections in the reference citations.
References
Alexander, J.W.: Topological invariants of knots and links. Trans. Amer. Math. Soc. 30(2), 275–306 (1928). https://doi.org/10.2307/1989123
Andruskiewitsch, N., Graña, M.: From racks to pointed Hopf algebras. Adv. Math. 178(2), 177–243 (2003). https://doi.org/10.1016/S0001-8708(02)00071-3
Carter, J.S., Jelsovsky, D., Kamada, S., Langford, L., Saito, M.: Quandle cohomology and state-sum invariants of knotted curves and surfaces. Trans. Amer. Math. Soc. 355(10), 3947–3989 (2003). https://doi.org/10.1090/S0002-9947-03-03046-0
Carter, S., Ishii, A., Saito, M., Tanaka, K.: Homology for quandles with partial group operations. Pacific J. Math. 287(1), 19–48 (2017). https://doi.org/10.2140/pjm.2017.287.19
Ishii, A.: Moves and invariants for knotted handlebodies. Algebr. Geom. Topol. 8(3), 1403–1418 (2008). https://doi.org/10.2140/agt.2008.8.1403
Ishii, A.: A multiple conjugation quandle and handlebody-knots. Topology Appl. 196(part B), 492–500 (2015). https://doi.org/10.1016/j.topol.2015.05.029
Ishii, A.: The fundamental multiple conjugation quandle of a handlebody-link. J. Math. Soc. Japan 74(1), 1–23 (2022). https://doi.org/10.2969/jmsj/84308430
Ishii, A., Murao, T.: Derivatives with MCQ Alexander pairs for handlebody-knots. preprint (2022)
Ishii, A., Oshiro, K.: Derivatives with Alexander pairs for quandles. to appear in Fund. Math. (2022)
Ishii, A., Iwakiri, M., Jang, Y., Oshiro, K.: A $G$-family of quandles and handlebody-knots. Illinois J. Math. 57(3), 817–838 (2013)
Ishii, A., Nikkuni, R., Oshiro, K.: On calculations of the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links. Osaka J. Math. 55(2), 297–313 (2018)
Joyce, D.: A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1), 37–65 (1982). https://doi.org/10.1016/0022-4049(82)90077-9
Lin, X.S.: Representations of knot groups and twisted Alexander polynomials. Acta Math. Sin.(Engl. Ser.) 17(3), 361–380 (2001). https://doi.org/10.1007/s101140100122
Matveev, S.V.: Distributive groupoids in knot theory. Mat. Sb. (N.S.) 119(161)(1), 78–88160 (1982)
Murao, T.: Linear extensions of multiple conjugation quandles and MCQ Alexander pairs. J. Algebra Appl. 20(3), 2150045–19 (2021). https://doi.org/10.1142/S0219498821500456
Murao, T.: Affine extensions of multiple conjugation quandles and augmented MCQ Alexander pairs. Topology Appl. 301, 107531–20 (2021). https://doi.org/10.1016/j.topol.2020.107531
Wada, M.: Twisted Alexander polynomial for finitely presentable groups. Topology 33(2), 241–256 (1994). https://doi.org/10.1016/0040-9383(94)90013-2
Acknowledgements
The author would like to thank Atsushi Ishii for his helpful comments. This work was supported by JSPS KAKENHI Grant Numbers JP20K22312 and JP21K13796.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Murao, T. On sufficiency of the definition of MCQ Alexander pairs in terms of invariants for handlebody-knots. Beitr Algebra Geom 64, 689–719 (2023). https://doi.org/10.1007/s13366-022-00652-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-022-00652-0
Keywords
- Multiple conjugation quandle
- Handlebody-knot
- f-twisted Alexander matrix
- MCQ Alexander pair
- Cohomologous