Connected quandles associated with pointed abelian groups

W. Edwin Clark, Mohamed Elhamdadi, Xiang-dong Hou, Masahico Saito and Timothy Yeatman

Vol. 264 (2013), No. 1, 31–60

DOI: 10.2140/pjm.2013.264.31

Abstract

A quandle is a self-distributive algebraic structure that appears in quasigroup and knot theories. For each abelian group A and c ∈ A, we define a quandle G(A,c) on ℤ₃ × A. These quandles are generalizations of a class of nonmedial 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 nonmedial 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.

Keywords

quandles, pointed abelian groups, knot colorings

Mathematical Subject Classification 2010

Primary: 57M25

Milestones

Received: 13 April 2012

Revised: 13 July 2012

Accepted: 16 July 2012

Published: 5 July 2013