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Quandle module quivers. (English) Zbl 1473.57025

Quandles are algebraic structures with axioms motivated by the Reidemeister moves from knot theory. Given a set of endomorphisms \(f \in \text{End}(X)\) of a finite quandle \(X,\) a quiver-valued invariant \(\mathcal{Q}_X(L)\) of oriented knots and links \(L\) is defined in [K. Cho and S. Nelson, J. Knot Theory Ramifications 28, No. 1, Article ID 1950001, 12 p. (2019; Zbl 1420.57032)] where the authors categorify the quandle counting inavariant. In [N. Andruskiewitsch and M. Graña, Adv. Math. 178, No. 2, 177–243 (2003; Zbl 1032.16028)] the notion of quandle modules was introduced. Further developing and connecting these concepts the present authors define quandle module quivers. As a result they obtain a new two-variable polynomial that can distinguish links with the same quandle module polynomial value. The authors provide ample examples in Section 3. At the end they leave some open problems for future research.

MSC:

57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)
20N02 Sets with a single binary operation (groupoids)

Software:

Knot Atlas
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Full Text: DOI arXiv

References:

[1] Andruskiewitsch, N. and Graña, M., From racks to pointed Hopf algebras, Adv. Math.178(2) (2003) 177-243. · Zbl 1032.16028
[2] D. Bar-Natan, The knot atlas, http://katlas.org/wiki/Main_Page. · Zbl 0898.57001
[3] Carter, J. S., Elhamdadi, M., Graña, M. and Saito, M., Cocycle knot invariants from quandle modules and generalized quandle homology, Osaka J. Math.42(3) (2005) 499-541. · Zbl 1089.57008
[4] Cho, K. and Nelson, S., Quandle cocycle quivers, Topology Appl.268 (2019), Article ID: 106908, 10 p. · Zbl 1473.57007
[5] Cho, K. and Nelson, S., Quandle coloring quivers, J. Knot Theory Ramifications28(1) (2019), Article ID: 1950001, 12 p. · Zbl 1420.57032
[6] Elhamdadi, M. and Nelson, S., Quandles—An Introduction to the Algebra of Knots, , Vol. 74 (American Mathematical Society, Providence, RI, 2015). · Zbl 1332.57007
[7] Haas, A., Heckel, G., Nelson, S., Yuen, J. and Zhang, Q., Rack module enhancements of counting invariants, Osaka J. Math.49(2) (2012) 471-488. · Zbl 1245.57009
[8] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra23(1) (1982) 37-65. · Zbl 0474.57003
[9] Matveev, S. V., Distributive groupoids in knot theory, Mat. Sb. (N.S.)119(161)(1) (1982) 78-88.
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