Connected Quandles Associated with Pointed Abelian Groups

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 Ga...

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Bibliographic Details
Published in:arXiv.org 2011-07
Main Authors: Clark, W Edwin, Elhamdadi, Mohamed, Xiang-dong, Hou, Saito, Masahico, Yeatman, Timothy
Format: Article
Language:English
Subjects:
Group theory
Isomorphism
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1107.5777