Links with finite n–quandles

Hoste, Jim; Shanahan, Patrick

doi:10.2140/agt.2017.17.2807

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Links with finite $n$–quandles

Jim Hoste and Patrick D Shanahan

Algebraic & Geometric Topology 17 (2017) 2807–2823

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 $n$ .

Keywords

quandle, branched cover, n-quandle, knot, link

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 14 July 2016

Revised: 10 March 2017

Accepted: 4 April 2017

Published: 19 September 2017

Authors

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

Volume 17, issue 5 (2017)