I think the author has confused NP with NP-hard (or NP-complete). Indeed, it is extremely unlikely that the unknottedness problem is NP-complete, as it is alrea

I think the author has confused NP with NP-hard (or NP-complete). Indeed, it is extremely unlikely that the unknottedness problem is NP-complete, as it is already in NP ∩ co-NP (assuming the generalized Riemann hypothesis). Furthermore, we know that the problem of computing many knot invariants is "only" BQP-complete, so it's possible that unknottedness is in BQP (i.e., efficiently solvable by quantum computers), though I don't know the expert opinion on this.