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.

[1]  Kenneth A. Perko,et al.  On the classification of knots , 1974 .

[2]  Richard Henderson,et al.  Symbolic computation with finite quandles , 2006, J. Symb. Comput..

[3]  E. Brieskorn,et al.  Automorphic sets and braids and singularities , 1988 .

[4]  Robert Sedgewick,et al.  Permutation Generation Methods , 1977, CSUR.

[5]  David Joyce,et al.  A classifying invariant of knots, the knot quandle , 1982 .

[6]  Sam Nelson.  Classification of Finite Alexander Quandles , 2002, math/0202281.

[7]  L. Kauffman,et al.  Biquandles and virtual links , 2004 .

[8]  Louis H. Kauffman.  Virtual Knot Theory , 1999, Eur. J. Comb..

[9]  Pedro Lopes,et al.  On Finite Racks and Quandles , 2004, math/0412487.

[10]  Roger Fenn,et al.  RACKS AND LINKS IN CODIMENSION TWO , 1992 .

[11]  H. Ryder.  The structure of racks , 1993 .

[12]  Masahico Saito,et al.  Surfaces in 4-Space , 2004 .

[13]  Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles , 2002, math/0206255.

[14]  S. Matveev.  DISTRIBUTIVE GROUPOIDS IN KNOT THEORY , 1984 .

[15]  R. Fenn,et al.  The equation [B,(A-1)(A,B)] = 0 and virtual knots and links , 2004 .

[16]  Masahico Saito,et al.  Quandle cohomology and state-sum invariants of knotted curves and surfaces , 1999, math/9903135.

[17]  Sam Nelson,et al.  MATRICES AND FINITE ALEXANDER QUANDLES , 2005, math/0509365.

[18]  Indecomposable racks of order p^2 , 2002, math/0203157.

[19]  Sam Nelson,et al.  Matrices and finite biquandles , 2005 .

[20]  S. Kamada,et al.  ABSTRACT LINK DIAGRAMS AND VIRTUAL KNOTS , 2000 .

[21]  Pedro Lopes,et al.  Quandles at Finite Temperatures II , 2002 .

[22]  Pedro Lopes.  Quandles at Finite Temperatures I , 2002 .

[23]  S. Satoh,et al.  A NOTE ON NON-CLASSICAL VIRTUAL KNOTS , 2004 .

[24]  Pedro Lopes,et al.  QUANDLES AT FINITE TEMPERATURES III , 2003 .

[25]  Vassily Olegovich Manturov,et al.  Virtual Biquandles , 2004 .

[26]  Sam Nelson,et al.  Matrices and Finite Quandles , 2004 .