Classification of Finite Alexander Quandles
Geometric Topology
2007-05-23 v2 Algebraic Topology
Abstract
Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific conditions on when two linear quandles are dual and which linear quandles are connected. We apply this result to obtain a procedure for classifying Alexander quandles of any finite order and as an application we list the numbers of distinct and connected Alexander quandles with up to fifteen elements.
Cite
@article{arxiv.math/0202281,
title = {Classification of Finite Alexander Quandles},
author = {Sam Nelson},
journal= {arXiv preprint arXiv:math/0202281},
year = {2007}
}
Comments
10 pages, LaTeX. Typos corrected, proof of Theorem 2.1 fixed. To appear in Topology Proceedings