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Links with finite $n$–quandles

Jim Hoste and Patrick D Shanahan

Algebraic & Geometric Topology 17 (2017) 2807–2823

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.

quandle, branched cover, n-quandle, knot, link
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Received: 14 July 2016
Revised: 10 March 2017
Accepted: 4 April 2017
Published: 19 September 2017
Jim Hoste
Department of Mathematics
Pitzer College
Claremont, CA
United States
Patrick D Shanahan
Department of Mathematics
Loyola Marymount University
Los Angeles, CA
United States