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Generalized Quandle Polynomials

Published online by Cambridge University Press:  20 November 2018

Sam Nelson
Sam Nelson*
Affiliation:
Department of Mathematical Sciences, Claremont McKenna College, Claremont, CA 91711, U.S.A.e-mail: knots@esotericka.org
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Abstract

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We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants that further generalize the quandle counting invariant.

Keywords

57M2776D99finite quandlesfinite biquandleslink invariants
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 147 - 158
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Carter, J. S., Elhamdadi, M., Graña, M., and Saito, M., Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math. 42(2005), no. 3, 499–541.Google Scholar
[2] 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), no. 10, 3947–3989. doi:10.1090/S0002-9947-03-03046-0Google Scholar
[3] Eisermann, M.. Quandle coverings and their Galois correspondence. http://arxiv.org/abs/math/0612459.Google Scholar
[4] Fenn, R. and Rourke, C., Racks and links in codimension two. J. Knot Theory Ramifications 1(1992), no. 4, 343–406. doi:10.1142/S0218216592000203Google Scholar
[5] Henderson, R., Macedo, T., and Nelson, S., Symbolic computation with finite quandles. J. Symbolic Comput. 41(2006) 811–817. doi:10.1016/j.jsc.2006.03.002Google Scholar
[6] Ho, B. and Nelson, S., Matrices and finite quandles. Homology Homotopy Appl. 7(2005), no. 1, 197–208.Google Scholar
[7] Joyce, D., A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23(1982), no. 1, 37–65. doi:10.1016/0022-4049(82)90077-9Google Scholar
[8] Kauffman, L. H. and Radford, D., Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links. In: Diagrammatic morphisms and applications (San Francisco, CA, 2000), Contemp. Math., 318, American Mathematical Society, Providence, RI, 2003, pp. 113–140.Google Scholar
[9] Lam, D. and Nelson, S., An isomorphism theorem for Alexander biquandles. Internat. J. Math. 20(2009), no. 1, 97–107.Google Scholar
[10] Matveev, S. V., Distributive groupoids in knot theory. (Russian) Mat. Sb. (N.S.) 119(161)(1982), no. 1, 78–88, 160.Google Scholar
[11] Nelson, S., A polynomial invariant of finite quandles. J. Algebra Appl. 7(2008), no. 2, 263–273. doi:10.1142/S0219498808002801Google Scholar
[12] Nelson, S. and Neumann, W., The 2-generalized knot group determines the knot. Commun. Contemp. Math. 10(2008), suppl. 1, 843–847. doi:10.1142/S0219199708003058Google Scholar
[13] Nelson, S. and Vo, J., Matrices and finite biquandles. Homology, Homotopy Appl. 8(2006), no. 2, 51–73.Google Scholar
[14] Tuffley, C., Generalised knot groups distinguish the square and granny knots. J. Knot Theory Ramifications 18(2009), no. 8, 1129–1157. doi:10.1142/S0218216509007385Google Scholar
[15] Zablow, J.. Intersections of curves on surfaces with disk families in handlebodies. J. Knot Theory Ramifications 15(2006), no. 5, 631–649. doi:10.1142/S0218216506004671Google Scholar