The Quandary of Quandles: The Borel Completeness of a Knot Invariant
City Tech - CUNY
We show that the isomorphism problems for left distributive algebras, racks, quandles, and keis are as complex as possible in the sense of Borel reducibility. These various kinds of algebraic structure are important for their connections with the theory of knots, links and braids, and in particular, Joyce showed that quandles could be used as complete invariants for tame knots. However, quandles have heuristically seemed to be unsatisfactory invariants. Our result confirms this view, showing that from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a harder problem.
Sheila Miller is an Assistant Professor at the New York City College of Technology, in CUNY. She received her Ph.D. from the University of Colorado at Boulder in 2007, with a thesis entitled “Free Left-Distributive Algebras,” written under the supervision of Richard Laver, and subsequently held a postdoctoral position at the USMA in West Point. In addition to set theory and distributive algebras, she studies mathematical biology, with ongoing research into populations of sea turtles.