#### Volume 17, issue 5 (2017)

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Links with finite $n$–quandles

### Jim Hoste and Patrick D Shanahan

Algebraic & Geometric Topology 17 (2017) 2807–2823
##### 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
Primary: 57M25
Secondary: 57M27