Matrices and finite biquandles

Abstract

We describe a way of representing finite biquandles with n elements 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 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.

Virtual Knot Invariants From Group Biquandles and Their Cocycles

“…It is interesting to compare these to the conditions for a birack [3,19,25]. The conditions T, M, B are called Wada's conditions.…”

Biquandle invariants for links in the projective space

“…The other types of the fifth Reidemeister move are compositions of the already considered type and the fourth Reidemeister move, so it is not necessary to consider them. The link diagram admissible coloring number in the projective space RP 3 is a generalization of the classical node invariant in the 3-dimensional sphere S 3 , which in its turn coincides with the classical node diagram admissible coloring number by the biquandle elements [1,3]. This generalization is as follows: We say that the link in the projective space is local if there exists a 3-dimensional ball in RP 3 comprising such a link.…”

Biquandle cohomology and state-sum invariants of links and surface-links

“…Two biquandles X and Y are said to be isomorphic if there is a biquandle isomorphism f : X → Y . In [19], S. Nelson and J. Vo classified biquandles of order 2, 3, and 4 up to biquandle isomorphism. There are 2 biquandles of order 2, 10 biquandles of order 3, and 57 non-quandle biquandles of order 4.…”