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Algebraic properties of quandle extensions and values of cocycle knot invariants. (English) Zbl 1361.57005

As a generalization of the quandle coloring invariant, the quandle cocycle invariant was introduced by J. S. Carter et al. in [Trans. Am. Math. Soc. 355, No. 10, 3947–3989 (2003; Zbl 1028.57003)]. This paper investigates when this cocycle invariant reduces to the quandle coloring invariant. More precisely, the authors prove that when the extension of the chosen finite quandle satisfies certain algebraic conditions, then the cocycle invariant reduces to the coloring invariant. One special case is: when the extension is a conjugation quandle then it is constant. In the end of this paper, one criterion for a connected quandle to be a conjugation quandle is provided. As an application, it is proved that for some quandles in the list of Rig quandles, the cocycle invariant can not provide more information than the quandle coloring invariant, although they have nontrivial second cohomology groups.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)

Citations:

Zbl 1028.57003

Software:

GitHub; RiG
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Full Text: DOI arXiv

References:

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