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2017 Links with finite $n$–quandles
Jim Hoste, Patrick Shanahan
Algebr. Geom. Topol. 17(5): 2807-2823 (2017). DOI: 10.2140/agt.2017.17.2807

Abstract

Associated to every oriented link L in the 3–sphere is its fundamental quandle and, for each n > 1, there is a certain quotient of the fundamental quandle called the n–quandle of the link. We prove a conjecture of Przytycki which asserts that the n–quandle of an oriented link L in the 3–sphere is finite if and only if the fundamental group of the n–fold cyclic branched cover of the 3–sphere, branched over L, is finite. We do this by extending into the setting of n–quandles, Joyce’s result that the fundamental quandle of a knot is isomorphic to a quandle whose elements are the cosets of the peripheral subgroup of the knot group. In addition to proving the conjecture, this relationship allows us to use the well-known Todd–Coxeter process to both enumerate the elements and find a multiplication table of a finite n–quandle of a link. We conclude the paper by using Dunbar’s classification of spherical 3–orbifolds to determine all links in the 3–sphere with a finite n–quandle for some n.

Citation

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Jim Hoste. Patrick Shanahan. "Links with finite $n$–quandles." Algebr. Geom. Topol. 17 (5) 2807 - 2823, 2017. https://doi.org/10.2140/agt.2017.17.2807

Information

Received: 14 July 2016; Revised: 10 March 2017; Accepted: 4 April 2017; Published: 2017
First available in Project Euclid: 16 November 2017

zbMATH: 06791384
MathSciNet: MR3704243
Digital Object Identifier: 10.2140/agt.2017.17.2807

Subjects:
Primary: 57M25
Secondary: 57M27

Rights: Copyright © 2017 Mathematical Sciences Publishers

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