Link homotopy invariant quandles

James R. Hughes, Elizabethtown College


We consider several approaches to defining a link homotopy version of the fundamental quandle Q(L) of a link L in S . We first define the reduced fundamental quandle RQ(L) as a quotient of Q(L). We show that RQ(L) is a link homotopy invariant that carries at least as much information as the meridian-preserving isomorphism class of Milnor's reduced group RG(L). We then show that operator reduction, a plausible alternative approach to defining RQ(L), fails to yield a link homotopy invariant. Finally, we give a geometric characterization of RQ(L), and offer a caveat regarding a seemingly simpler approach. © 2011 World Scientific Publishing Company. 3