An enumeration process for racks

TL;DR

This paper introduces a systematic enumeration process for racks, generalizing existing methods, with proven termination criteria and applications to knot theory, enhancing the computational tools for algebraic structures in topology.

Contribution

It develops a new enumeration process for racks based on Todd-Coxeter, improving diagramming methods and providing termination proofs and pseudocode.

Findings

Process terminates iff the rack is finite.

Outputs an operation table for finite racks.

Application demonstrated in knot theory.

Abstract

Given a presentation for a rack $\mathcal R$, we define a process which systematically enumerates the elements of $\mathcal R$. The process is modeled on the systematic enumeration of cosets first given by Todd and Coxeter. This generalizes and improves the diagramming method for $n$-quandles introduced by Winker. We provide pseudocode that is similar to that given by Holt for the Todd-Coxeter process. We prove that the process terminates if and only if $\mathcal R$ is finite, in which case, the procedure outputs an operation table for the finite rack. We conclude with an application to knot theory.