Galkin Quandles, Pointed Abelian Groups, and Sequence A000712

2011/08/10 by W. Edwin Clark, Clark, W. Edwin, Xiang‐dong Hou +1 · 1 citation
Computer Science · Mathematics · #05A17 #20K30 #57M27 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1108.2215

openalex publication_date 2011/08/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each pointed abelian group (A,c), there is an associated \em Galkin quandle G(A,c) which is an algebraic structure defined on \Bbb Z₃× A 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 qⁿ (q prime) is ∑_{0≤ m≤ n}p(m)p(n-m), where p(m) is the number of partitions of integer m.

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