Matrices and finite quandles

Abstract

Finite quandles with n elements can be represented as n × n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.

On the Orbit Decomposition of Finite Quandles

“…The lists of quandles of order n ≤ 6 in [2] and order n ≤ 5 in [9] show that many of the possible quandle structures of small order may be understood as unions of disjoint subquandles. This observation naturally raises the question of how many different ways there are for two (or more) quandles to be put together, that is, how many quandle structures are possible on Q ∐ Q ′ which have Q and Q ′ as subquandles.…”

“…In section 3, we use quandle matrices (see [9]) to study the structure of finite quandles. We describe algorithms for finding the orbit decomposition of a finite quandle and for finding quandle structures on a disjoint union of n subquandles.…”

Quandles and Linking Number

“…As noted in [3], the first two quandle axioms are easy to check since the diagonal entries are 1, 2, 3, . .…”

Matrices and finite biquandles

“…In order to find and evaluate such invariants, then, we need a way of representing finite biquandles. In [8], finite quandles Q = {x 1 , . .…”

“…In this paper, we study finite biquandles using a matrix notation similar to the quandle matrix notation in [8]; these finite biquandles define invariants of virtual and classical knots via homomorphism counting much like finite groups and finite quandles. The paper is organized as follows: in Section 2, we give the biquandle definition and prove some results about finite biquandles.…”