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Preprints, Working Papers, ... Year : 2019

Abelian quandles and quandles with abelian structure group

Abstract

Sets with a self-distributive operation (in the sense of (a ⊳ b) ⊳ c = (a ⊳ c) ⊳ (b ⊳ c)), in particular quandles, appear in knot and braid theories, Hopf algebra classification, the study of the Yang-Baxter equation, and other areas. An important invariant of quandles is their structure group. The structure group of a finite quandle is known to be either "boring" (free abelian), or "interesting" (non-abelian with torsion). In this paper we explicitly describe all finite quandles with abelian structure group. To achieve this, we show that such quandles are abelian (i.e., satisfy (a ⊳ b) ⊳ c = (a ⊳ c) ⊳ b); present the structure group of any abelian quandle as a central extension of a free abelian group by an explicit finite abelian group; and determine when the latter is trivial. In the second part of the paper, we relate the structure group of any quandle to its 2nd homology group H 2. We use this to prove that the H 2 of a finite quandle with abelian structure group is torsion-free, but general abelian quandles may exhibit torsion. Torsion in H 2 is important for constructing knot invariants and pointed Hopf algebras.
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Dates and versions

hal-02266880 , version 1 (16-08-2019)
hal-02266880 , version 2 (20-08-2019)
hal-02266880 , version 3 (02-09-2019)

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Victoria Lebed, Arnaud Mortier. Abelian quandles and quandles with abelian structure group. 2019. ⟨hal-02266880v3⟩
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