On the Classification of Topological Quandles

TL;DR

This paper classifies topological quandle structures on simple manifolds like the real line and circle, and conjectures uniqueness of the trivial structure on the interval, providing supporting evidence.

Contribution

It provides a complete classification of Alexander quandle structures on simple manifolds and proposes a conjecture on the uniqueness of the trivial structure on the interval.

Findings

Classified Alexander quandle structures on the real line and circle.

Conjectured the trivial structure is unique on the interval.

Provided evidence supporting the conjecture.

Abstract

We investigate the classification of topological quandles on some simple manifolds. Precisely we classify all Alexander quandle structures, up to isomorphism, on the real line and the unit circle. For the closed unit interval $[0, 1]$, we conjecture that there exists only one topological quandle structure on it, i.e. the trivial one. Some evidences are provided to support our conjecture.