A homogeneous presentation of symmetric quandles

TL;DR

This paper introduces a new construction method for symmetric quandles using groups and subgroups, and proves that all symmetric quandles can be represented as disjoint unions of these structures.

Contribution

It provides a systematic way to construct symmetric quandles from groups and shows that all symmetric quandles are isomorphic to these constructions.

Findings

Constructed symmetric quandles from groups and subgroups.

Proved all symmetric quandles are isomorphic to the constructed form.

Abstract

A quandle is an algebraic structure whose axioms correspond to the Reidemeister moves of knot theory. S. Kamada introduced the notion of a quandle with a good involution, which is later called a symmetric quandle. We are interested in the algebraic structure of symmetric quandles. Given a group $G$, an element $z$ and a certain subgroup $H$, one can obtain the quandle. D. Joyce showed that every quandle is isomorphic to the disjoint union of such quandles. In this paper, given a group $G$, elements $z,r$ in $G$ and a certain subgroup $H$, we construct a symmetric quandle. Futhermore, we show that every symmetric quandle is isomorphic to the disjoint union of such quandles.