Note on spherical quandles

TL;DR

This paper establishes a correspondence between $SU(2)$-representations of knot groups and spherical quandle colorings, providing a geometric interpretation of the trace-free condition used in defining the Casson-Lin invariant.

Contribution

It introduces a novel link between knot group representations and spherical quandle colorings, clarifying the geometric meaning of the trace-free condition in knot invariants.

Findings

Established a one-to-one correspondence between $SU(2)$-representations and spherical quandle colorings.

Provided geometric insight into the trace-free condition for the Casson-Lin invariant.

Enhanced understanding of knot invariants through quandle theory.

Abstract

This paper aims to give a one-to-one correspondence between $SU(2)$-representations of knot groups and colorings of knots with spherical quandles and give a geometric meaning of the "trace-free" condition we need to define Casson-Lin invariant.