Vol. 264, No. 1, 2013

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ISSN: 0030-8730
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
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 3 × 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
Authors
W. Edwin Clark
Department of Mathematics
University of South Florida
Tampa, FL 33620-5700
United States
Mohamed Elhamdadi
Department of Mathematics
University of South Florida
Tampa, FL 33620-5700
United States
Xiang-dong Hou
Department of Mathematics and Statistics
University of South Florida
Tampa, FL 33620-5700
United States
Masahico Saito
Department of Mathematics and Statistics
University of South Florida
Tampa, FL 33620-5700
United States
Timothy Yeatman
Department of Mathematics and Statistics
University of South Florida
Tampa, FL 33620-5700
United States