Detecting knot invertibility

Abstract

We discuss the consequences of the possibility that Vassiliev invariants do not detect knot invertibility as well as the fact that quantum Lie group invariants are known not to do so. On the other hand, finite group invariants, such as the set of homomorphisms from the knot group to M_11, can detect knot invertibility. For many natural classes of knot invariants, including Vassiliev invariants and quantum Lie group invariants, we can conclude that the invariants either distinguish all oriented knots, or there exist prime, unoriented knots which they do not distinguish.