Quandles versus symmetric quandles for oriented links

TL;DR

This paper explores the relationship between quandles and symmetric quandles, demonstrating their (co)homology equivalence and showing how symmetric quandles can be used to interpret link invariants.

Contribution

It introduces the symmetric double of a quandle and proves the isomorphism of (co)homology groups, extending the application of symmetric quandles to link invariants.

Findings

(co)homology groups of a quandle and its symmetric double are isomorphic

Symmetric quandles can interpret quandle coloring numbers and cocycle invariants

The symmetric double construction preserves key algebraic properties

Abstract

Given a quandle, we can construct a symmetric quandle called the symmetric double of the quandle. We show that the (co)homology groups of a given quandle are isomorphic to those of its symmetric double. Moreover, quandle coloring numbers and quandle cocycle invariants of oriented links and oriented surface-links can be interpreted by using symmetric quandles.