Connected Quandles Associated with Pointed Abelian Groups

TL;DR

This paper introduces Galkin quandles, a new class of connected quandles constructed from pointed abelian groups, classifies their isomorphism classes, and explores their properties and applications in knot theory.

Contribution

It defines Galkin quandles based on abelian groups, classifies their isomorphism classes, and investigates their algebraic properties and relevance to knot colorings.

Findings

Galkin quandles are connected but not Latin unless A has odd order.

They are non-medial unless 3A=0.

A classification of isomorphism classes in terms of pointed abelian groups.

Abstract

A quandle is a self-distributive algebraic structure that appears in quasi-group and knot theories. For each abelian group A and c \in A we define a quandle G(A, c) on \Z_3 \times A. These quandles are generalizations of a class of non-medial Latin quandles defined by V. M. Galkin so we call them Galkin quandles. Each G(A, c) is connected but not Latin unless A has odd order. G(A, c) is non-medial unless 3A = 0. We classify their isomorphism classes in terms of pointed abelian groups, and study their various properties. A family of symmetric connected quandles is constructed from Galkin quandles, and some aspects of knot colorings by Galkin quandles are also discussed.