  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.
