Narrow Results

We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe an algorithm for finding quandle structures on a disjoint union of quandles.

We study the question of whether a finite quandle specified by a matrix is isomorphic to an Alexander quandle. We first collect some necessary conditions for a finite quandle to be Alexander. We then describe an algorithm for determining finding all possible Alexander presentations of a finite quandle given its matrix and provide a URL for Maple code implementing this algorithm.

We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

We prove that an Alexander quandle of prime order is generated by any pair of distinct elements. Furthermore, we prove for such a quandle that any ordered pair of distinct elements can be sent to any other such pair by an automorphism of the quandle.

We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.