Finite involutory quandles of two-bridge links with an axis

TL;DR

This paper explicitly describes the Cayley graphs of finite involutory quandles associated with two-bridge links with an axis, advancing understanding of their algebraic structure.

Contribution

It provides explicit descriptions of the Cayley graphs for finite involutory quandles of two-bridge links with an axis, extending previous classifications.

Findings

Explicit Cayley graphs for these quandles are constructed.

The work classifies finite involutory quandles for a specific class of links.

Enhances understanding of the algebraic structure of link quandles.

Abstract

To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the {\em involutory} quandle). Hoste and Shanahan \cite{HS2} gave a complete list of the links which have finite $n$-quandles; it remained to give explicit descriptions of these quandles. This has been done for several cases in \cite{CHMS} and \cite{HS1}; in the current work we continue this project and explicitly describe the Cayley graphs for the finite involutory quandles of two-bridge links with an axis.