The Topological Quandles up to Four Elements

TL;DR

This paper classifies all finite topological quandles with up to four elements using matrix representations, providing a comprehensive understanding of their isomorphism classes.

Contribution

It introduces a method to distinguish finite topological quandles via matrices and completes the classification for quandles with up to four elements.

Findings

Complete classification of finite topological quandles with up to 4 elements.

Matrix-based method to distinguish isomorphism classes.

Identification of all isomorphism classes for small cardinalities.

Abstract

The finite topological quandles can be represented as $n\times n$ matrices, recently defined by S. Nelson and C. Wong. In this paper, we first study the finite topological quandles and we show how to use these matrices to distinguish all isomorphism classes of finite topological quandles for a given cardinality $n$. As an application, we classify finite topological quandles with up to 4 elements.