I found it somewhat difficult to understand the status of certain problems from knot theory. Is it correct to say that it's been neither proved nor disproved that any of the following problems are NP-complete:
- The equivalence problem for links (and knots) as described in knot theory.
- The unknot recognition problem i.e. to determine if an arbitrary knot is trivial (the empty knot).
- Computing the link crossing number for an arbitrary link (or knot).
But it is known that the above problems are decidable and some (all?) of them are NP-hard. I thought I read something that the computation of the homfly polynomial is NP complete via a reduction to colouring of graphs, but can we confirm that it is the case?