Links with finite $n$–quandles

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 $n$.