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

 The Computational Complexity of Knot Problems. 
Knottedness is in NP, modulo GRH. .
Unknotting is in $\mathrm{AM}\cap\mathrm{co\text-AM} $.

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?
Such a quasi-polynomial algorithm has just been claimed by Marc Lackenby from Oxford University. He will present in next Tuesday (02 Feb 2021) in a Zoom talk: https://www.math.ucdavis.edu/research/seminars/?talk_id=6082