N-quandles of links

TL;DR

This paper introduces N-quandles, a generalization of n-quandles, and explores their finiteness properties for links, providing partial classification results and conjectures in knot theory.

Contribution

It generalizes n-quandles to N-quandles for links and offers initial classification results along with conjectures about their finiteness.

Findings

Identified conditions under which N-quandles are finite.

Proved one direction of the conjectured classification.

Extended the framework of quandle invariants in knot theory.

Abstract

The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is the $n$-quandle. Hoste and Shanahan gave a complete list of the knots and links which have finite $n$-quandles for some $n$. We introduce a generalization of $n$-quandles, denoted $N$-quandles (for a quandle with $k$ algebraic components, $N$ is a $k$-tuple of positive integers). We conjecture a classification of the links with finite $N$-quandles for some $N$, and we prove one direction of the classification.