Topology Seminar

Speaker: Jim Hoste, Pitzer College

Title: Links with finite n-quandles

Abstract: Associated to every knot is its fundamental quandle Q(K), which Joyce proved is a complete knot invariant. A somewhat more tractable, but less sensitive invariant is the n-quandle, a quotient of Q(K) defined for every natural number n. I will describe these quandles and show that the n-quandle of a knot is isomorphic to the set of cosets of the peripheral subgroup of a certain quotient of the fundamental group of the knot. This characterization proves a conjecture of Przytycki: The n-quandle of a knot is finite if and only if the fundamental group of the n-fold cyclic cover of S^3 branched over the knot is finite. I will outline a program to catalog all finite quandles that appear as n-quandles of some knot or link. Some of this is joint work with Pat Shanahan.