Homology, Homotopy and Applications

Volume 8 (2006)

Number 2

Matrices and finite biquandles

Pages: 51 – 73

DOI: https://dx.doi.org/10.4310/HHA.2006.v8.n2.a3


Sam Nelson (Department of Mathematics, University of California at Riverside)

John Vo (Department of Mathematics, University of California at Riverside)


We describe away of representing finite biquandles with $n$ elements as $2n \times 2n$ 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 nontriviality of the virtual trefoil and various Kishino knots. We also exhibit an oriented virtual knot which is distinguished from both its obverse and its reverse by a finite biquandle counting invariant. We classify biquandles of order 2, 3 and 4 and provide a URL for our Maple programs for computing with finite biquandles.


finite biquandles, virtual knot invariants

2010 Mathematics Subject Classification

57M25, 57M27

Published 1 January 2006