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Tribracket modules. (English) Zbl 1440.57008

Summary: Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper.

MSC:

57K10 Knot theory
57K12 Generalized knots (virtual knots, welded knots, quandles, etc.)

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] Bauernschmidt, R. and Nelson, S., Birack modules and their link invariants, Commun. Contemp. Math.15(3) (2013) 1350006, 13. · Zbl 1290.57018
[4] 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
[5] 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(10) (2003) 3947-3989. · Zbl 1028.57003
[6] Ceniceros, J. and Nelson, S., \((t,s)\)-racks and their link invariants, Internat. J. Math.23(3) (2012) 1250001, 19. · Zbl 1251.57013
[7] Cody, E. and Nelson, S., Polynomial birack modules, Topology Appl.173 (2014) 285-293. · Zbl 1298.57008
[8] Elhamdadi, M. and Nelson, S., Quandles — An Introduction to the Algebra of Knots, , Vol. 74 (American Mathematical Society, Providence, RI, 2015). · Zbl 1332.57007
[9] P. Graves, S. Nelson and S. Tamagawa, Niebrzydowski algebras and trivalent spatial graphs, preprint arXiv:1805.00104 (2018). · Zbl 1406.57010
[10] 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
[11] Joyce, D., A classifying invariant of knots, the knot quandle, J. Pure Appl. Algebra23(1) (1982) 37-65. · Zbl 0474.57003
[12] Kim, J. and Nelson, S., Biquasile colorings of oriented surface-links, Topology Appl.236 (2018) 64-76. · Zbl 1383.57031
[13] Matveev, S. V., Distributive groupoids in knot theory, Mat. Sb. (N.S.)119(161)(1) (1982) 78-88, 160.
[14] Needell, D. and Nelson, S., Biquasiles and dual graph diagrams, J. Knot Theory Ramifications26(8) (2017) 1750048, 18. · Zbl 1372.57030
[15] Nelson, S., Oshiro, K., and Oyamaguchi, N., Local biquandles and Niebrzydowski’s tribracket theory, Topology Appl.258 (2019) 474-512. · Zbl 1415.57007
[16] Nelson, S. and Pelland, K., Birack shadow modules and their link invariants, J. Knot Theory Ramifications22(10) (2013) 1350056, 12. · Zbl 1317.57011
[17] S. Nelson and S. Pico, Virtual tribrackets, preprint (2018), arXiv:1803.03210. · Zbl 1426.57029
[18] Niebrzydowski, M., On some ternary operations in knot theory, Fund. Math.225(1) (2014) 259-276. · Zbl 1294.57008
[19] M. Niebrzydowski, Homology of ternary algebras yielding invariants of knots and knotted surfaces, preprint (2017), arXiv:1706.04307. · Zbl 1464.57004
[20] Polyak, M., Minimal generating sets of Reidemeister moves, Quantum Topol.1(4) (2010) 399-411. · Zbl 1229.57012
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