11 research outputs found

    Symbolic computation with finite biquandles

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    A method of computing a basis for the second Yang-Baxter cohomology of a finite biquandle with coefficients in Q and Z_p from a matrix presentation of the finite biquandle is described. We also describe a method for computing the Yang-Baxter cocycle invariants of an oriented knot or link represented as a signed Gauss code. We provide a URL for our Maple implementations of these algorithms.Comment: 8 pages. Version 2 has typo corrections and changes suggested by referee. To appear in J. Symbolic Compu

    Generalized quandle polynomials

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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 which further generalize the quandle counting invariant.Comment: 11 pages. Version 3 includes a correction to the square/granny knot example. To appear in Can. Bull. Mat

    A Survey of Quandle Ideas

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    This article surveys many aspects of the theory of quandles which algebraically encode the Reidemeister moves. In addition to knot theory, quandles have found applications in other areas which are only mentioned in passing here. The main purpose is to give a short introduction to the subject and a guide to the applications that have been found thus far for quandle cocycle invariants.Comment: Submitted to conference proceedings; embarrassing misspellings of various names corrected. Many apologies and thanks to readers who pointed out correction

    A Survey of Racks and Quandles: Some recent developments

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    This short survey contains some recent developments of the algebraic theory of racks and quandles. We report on some elements of representation theory of quandles and ring theoretic approach to quandles.Comment: 13 pages. ICART 2018. To appear in Algebra Colloquiu

    Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs

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    Building on a result by W. Rump, we show how to exploit the right-cyclic law (x.y).(x.z) = (y.x).(y.z) in order to investigate the structure groups and monoids attached with (involutive nondegenerate) set-theoretic solutions of the Yang-Baxter equation. We develop a sort of right-cyclic calculus, and use it to obtain short proofs for the existence both of the Garside structure and of the I-structure of such groups. We describe finite quotients that exactly play for the considered groups the role that Coxeter groups play for Artin-Tits groups
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