Classification of Finite Alexander Quandles

Details Classification of Finite Alexander Quandles Classification of Finite Alexander Quandles

2002-02-26

arxiv.org/abs/math/0202281v2

Topology Proceedings 27 (2003) pp. 245-258.

Mathematics

Geometric Topology

10 pages, LaTeX. Typos corrected, proof of Theorem 2.1 fixed. To appear in Topology Proceedings

Scientific paper

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.

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