Matrices and finite biquandles

Wedescribeawayofrepresentingfinitebiquandleswithnelements as 2n ◊ 2n block matrices. Any finite biquandle defines an invariant of virtual knots through counting homomorphisms. The counting invariants of non-quandle biquandles can reveal information not present in the knot quandle, such as the nontrivialityofthevirtualtrefoilandvariousKishinoknots.Wealso exhibitanorientedvirtualknotwhichisdistinguishedfromboth its obverse and its reverse by a finite biquandle counting invariant.Weclassifybiquandlesoforder2,3and4andprovideaURL for our Maple programs for computing with finite biquandles.

[1]  Sam Nelson,et al.  Symbolic computation with finite biquandles , 2005, J. Symb. Comput..

[2]  Sam Nelson,et al.  Matrices and Finite Quandles , 2004, math/0412417.

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

[4]  L. Kauffman,et al.  Virtual Biquandles , 2004, math/0411243.

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

[6]  Sam Nelson.  Signed ordered knotlike quandle presentations , 2004, math/0408375.

[7]  V. Turaev,et al.  Problems on invariants of knots and 3-manifolds , 2004, math/0406190.

[8]  M. Saito,et al.  Generalizations of Quandle Cocycle Invariants and Alexander Modules from Quandle Modules , 2004, math/0401183.

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

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

[11]  D. Yetter.  Quandles and Monodromy , 2002, math/0205162.

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

[13]  L. Kauffman,et al.  Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links , 2001, math/0112280.

[14]  Ayumu Inoue.  QUANDLE HOMOMORPHISMS OF KNOT QUANDLES TO ALEXANDER QUANDLES , 2001 .

[15]  P. Lopes.  Quandles at Finite Temperatures I , 2001, math/0105099.

[16]  J. Sawollek.  On Alexander-Conway Polynomials for Virtual Knots and Links , 1999, math/9912173.

[17]  L. Kauffman.  Virtual Knot Theory , 1998, Eur. J. Comb..

[18]  Roger Fenn,et al.  Trunks and classifying spaces , 1995, Appl. Categorical Struct..

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

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

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