I was reading a little about knots (in a popular math book that wasn't very good) and the book put forth several knot invariants like the Alexander and Jones polynomials. But these are not complete invariants. This made me wonder "what are some complete knot invariants?" and the obvious next question "what is the descriptive complexity of knot equivalence as an analytic equivalence relation?"

Knot equivalence is an analytic equivalence relation on a Polish space (http://en.wikipedia.org/wiki/Knot_theory#Knot_equivalence) and there do exist complete knot invariants (Complete knot invariant?). But I didn't know if this equivalence relation had been looked at from the descriptive set theory point of view.