Automorphisms and antiautomorphisms of quandles

TL;DR

This paper characterizes when automorphisms and antiautomorphisms of groups induce similar structures on various types of quandles, and constructs new automorphisms not derived from the original group.

Contribution

It provides conditions for automorphisms of groups to induce automorphisms of associated quandles and constructs novel automorphisms independent of the group.

Findings

Conditions for automorphisms to induce quandle automorphisms

Construction of automorphisms not from the group

Analysis of automorphisms in different quandle types

Abstract

In this paper we provide the conditions under which an automorphism or an antiautomorphism of a group $G$ induces an automorphism or an antiautomorphism of the $m$-conjugation quandle $\operatorname{Conj_{m}}(G),\,\, m\in \mathbb{Z} $, the core quandle $\operatorname{Core}(G)$, the generalized Alexander quandle $\operatorname{Alex}(G,\phi)$ where $\phi\in \operatorname{Aut}(G)$ and some others. We also construct automorphisms of these quandles that do not originate from $G$.