Narrow Results

A (t, s)-rack is a rack structure defined on a module over the ring . We identify necessary and sufficient conditions for two (t, s)-racks to be isomorphic. We define enhancements of the rack counting invariant using the structure of (t, s)-racks and give some computations and examples. As an application, we use these enhanced invariants to obtain obstructions to knot ordering.

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.

Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.

Colorings of torus knots and twist-spun torus knots by general Alexander quandles over finite fields are explicitly determined. It is then observed that, for some standard Mochizuki's cocycles, a large class of twist-spun torus knots with non-trivial colorings have trivial CJKLS state-sum invariants.

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.