Galkin Quandles, Pointed Abelian Groups, and Sequence A000712 (original) (raw)

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Keywords: Galkin quandle, knot, Frobenius symbol, partition number, pointed abelian group

Abstract

For each pointed abelian group (A,c)(A,c)(A,c), there is an associated Galkin quandle G(A,c)G(A,c)G(A,c) which is an algebraic structure defined on BbbZ3timesA\Bbb Z_3\times ABbbZ3timesA that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order qnq^nqn ($q$ prime) is sum0lemlenp(m)p(n−m)\sum_{0\le m\le n}p(m)p(n-m)sum0lemlenp(m)p(n−m), where p(m)p(m)p(m) is the number of partitions of integer mmm.

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How to Cite

Clark, W. E., & Hou, X.- dong. (2013). Galkin Quandles, Pointed Abelian Groups, and Sequence A000712. The Electronic Journal of Combinatorics, 20(1), #P45. https://doi.org/10.37236/2676