# Complexity of Unknotting problems

The complexity of the Unknotting problem is known to be in $$\mathrm{NP} \cap\mathrm{co\text-NP}$$, see references:

Some even believe the problem to be in $$P$$, but to relax the ptime constraints, due to (maybe) either the cost, or diagram of the knot. It maybe there exists an Arthur-Merlin type protocol for unknottedness. This could lead to possibly interesting applications too. Also, if Unknottedness is $$\mathrm{NP}$$ -complete then the polynomial hierarchy collapses.

But by relaxing the ptime constraints to a quasipolynomial one leads one to quasipolynomial time algorithm for Unknotting. Is there any such quasipolynomial time algorithm for Unknotting?

What (if any) would be the implications of such a quasipolynomial algorithm be to other conjectures such as GRH(?), ETH, or even SETH?