TITLE: Descriptive Complexity of Knot Equivalence
QUESTION [14 upvotes]: 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.

REPLY [3 votes]: For this answer, we will restrict to the class of tame knots, i.e. knots that can be smoothly embedded in $S^3$.
Knot polynomials are often not complete invariants of knots and links. For example, there are several knots known to have trivial Alexander polynomial. While it is open if the unknot is detected by the Jones' polynomial, there are also knots known to have the same Jones' polynomial, for example the Kanenobu knots, named after the knots that appear in Theorem 4 the following paper:

Taizo Kanenobu, Examples on polynomial invariants of knots and links
  Mathematische Annalen. 1986, Volume 275, Issue 4, pp 555-572

Also, there are knots with the same HOMFLY-PT polynomial for example mutant knots, see:

W.B.R Lickorish, Linear skein theory and link polynomials, Topology and its
   Applications, 27 (1987), 265-274.

However, knots can be distinguished up to ambient isotopy by their complements, which is a famous (and non-trivial) theorem of Gordon and Luecke:

Cameron Gordon and John Luecke, Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), no. 2, 371–415.

It is worth mentioning that the analogous statement does not hold for links and link complements.  
In terms of descriptive complexity, many upper bounds on the descriptive complexity of distinguishing knots are now framed in terms of the descriptive complexity of  distinguishing a specific knot complement from another 3-manifold. Marc Lackenby's recent work along these lines shows that the unknot can be recognized in polynomial time:

Marc Lackenby, A polynomial upper bound on Reidemeister, to appear Annals of Math.
  Currently on the arXiv