Document Zbl 1479.57025 - zbMATH Open

While a quandle is a set with a binary operation satisfying axioms analogous to the Reidemeister moves of knot diagrams, a multiple conjugation quandle (MCQ) \(X\) is a disjoint union of groups \(\coprod_\lambda G_\lambda\) with a binary operation \(\triangleleft:X\times X\rightarrow X\) satisfying axioms corresponding to the Reidemeister moves of handlebody-knot diagrams [A. Ishii, Topology Appl. 196, Part B, 492–500 (2015; Zbl 1348.57024)].
An MCQ homomorphism \(f:X_1\rightarrow X_2\) is a map between two MCQ’s \(X_1,X_2\) that respects the operation \(\triangleleft\) and restricts to a homomorphism on each \(G_\lambda\). An MCQ \(\tilde{X}\) is called an extension of another MCQ \(X\) if there is a surjective MCQ homomorphism \(\pi:\tilde{X}\rightarrow X\) such that the cardinality of \(\pi^{-1}(x)\) is independent of \(x\in X\).
The paper under review introduces the notion of an MCQ Alexander pair \((f_1,f_2)\) for an MCQ \(X\) which are maps \(f_1,f_2\) from \(X\times X\) to a ring \(R\) under suitable algebraic conditions. An MCQ Alexander pair can be seen as an analog of an Alexander pair for quandles introduced in [A. Ishii and K. Oshiro, “Twisted derivatives with Alexander pairs for quandles”, Preprint, https://www.math.tsukuba.ac.jp/~aishii/files/paper035.pdf:2020]. The author shows that every linear extension of an MCQ associated to a quadruple of maps can be realized by an MCQ Alexander pair, up to isomorphism.

MSC:

57K12	Generalized knots (virtual knots, welded knots, quandles, etc.)
20N02	Sets with a single binary operation (groupoids)
57M15	Relations of low-dimensional topology with graph theory

References:

[1]	Alexander, J. W., Topological invariants of knots and links, Trans. Amer. Math. Soc.30 (1928) 275-306. · JFM 54.0603.03
[2]	Andruskiewitsch, N. and Graña, M., From racks to pointed Hopf algebras, Adv. Math.178(2) (2003) 177-243. · Zbl 1032.16028
[3]	Carter, J. S., Jelsovsky, D., Kamada, S., Langford, L. and Saito, M., Quandle cohomology and state-sum invariants of knotted curves and surfaces, Trans. Amer. Math. Soc.355 (2003) 3947-3989. · Zbl 1028.57003
[4]	M. Elhamdadi and S. Nelson, Quandles — An introduction to the algebra of knots, Student Mathematical Library, Vol. 74 (American Mathematical Society, Providence, RI, 2015). · Zbl 1332.57007
[5]	Ishii, A., Moves and invariants for knotted handlebodies, Algebr. Geom. Topol.8 (2008) 1403-1418. · Zbl 1151.57007
[6]	Ishii, A., A multiple conjugation quandle and handlebody-knots, Topology Appl.196 (2015) 492-500. · Zbl 1348.57024
[7]	Ishii, A., The Markov theorem for spatial graphs and handlebody-knots with Y-orientations, Internat. J. Math.26 (2015) 1550116, 23 pp. · Zbl 1337.57004
[8]	Ishii, A., Iwakiri, M., Jang, Y. and Oshiro, K., A \(G\)-family of quandles and handlebody-knots, Illinois J. Math.57 (2013) 817-838. · Zbl 1306.57011
[9]	A. Ishii and T. Murao, Twisted derivatives with MCQ Alexander pairs for multiple conjugation quandles, preprint.
[10]	A. Ishii and K. Oshiro, Twisted Derivatives with Alexander pairs for quandles, preprint.
[11]	Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra23 (1982) 37-65. · Zbl 0474.57003
[12]	Lin, X. S., Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.)17 (2001) 361-380. · Zbl 0986.57003
[13]	Matveev, S. V., Distributive groupoids in knot theory, Mat. Sb. (N.S.)119(161) (1982) 78-88. · Zbl 0523.57006
[14]	Nosaka, T., Quandles and Topological Pairs: Symmetry, Knots, and Cohomology, (Springer, Singapore, 2017). · Zbl 1411.57001
[15]	Wada, M., Twisted Alexander polynomial for finitely presentable groups, Topology33(2) (1994) 241-256. · Zbl 0822.57006

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