Symbolic computation with finite quandles

Algorithms are described and Maple implementations are provided for finding all quandles of order $n$, as well as computing all homomorphisms between two finite quandles or from a finitely presented quandle (e.g., a knot quandle) to a finite quandle, computing the automorphism group of a finite quandle, etc. Several of these programs work for arbitrary binary operation tables and hence algebraic structures other than quandles. We also include a stand-alone C program which finds quandles of order $n$ and provide URLs for files containing the results for $n=6,$ 7 and 8.Comment: 6 pages. Version 2 includes some updates and corrections. Code has been relocated to www.esotericka.org/quandles. To appear in J. Symbolic Computatio

Generalized quandle polynomials

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

Symbolic computation with finite biquandles

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