Andrew Brooke-Taylor (University of Leeds) – The complexity of quandles

We all know that quandles are great, but there seems to have been resistance to their use as knot invariants because they're too hard to work with.  In this talk I will explain an older result of mine with Sheila Miller, giving rigorous mathematical substance to this sociological observation: we show that, in the framework of "Borel reducibility", the isomorphism problem for quandles is as complicated as it could possibly be.  The proof boils down to a very hands-on, combinatorial construction of a quandle from any given graph.