TITLE: What part is left unsolved in the Unknotting problem? (after results of Bar-Natan, Khovanov, Kronheimer and Mrowka)
QUESTION [30 upvotes]: Kronheimer and Mrowka showed that the Khovanov homology detects the unknot.
Bar-Natan showed a program to compute the Khovanov homology fast: there was no rigorous complexity analysis of the algorithm, but it is estimated by Bar-Natan that the algorithm runs in time proportional to the square root of the number of crossings, so it is even less than linear in the number of crossings.

What I might understand from this, is that if we have:

a proof for the correctness of Bar-Natan's algorithm
a rigorous algorithm analysis showing the run time of the algorithm is polynomial or less

then we have a proof that the unknotting problem is in P.
I guess this is not really the case.
Is what I assume here true? if not, why? (maybe Bar-Natan's algorithm is not deterministic?)

REPLY [24 votes]: This isn't directly what you ask, but it's also worth noting that unknot detection is in $\text{NP} \cap \text{co-NP}$, that is, there are polynomial-checkable certificates that will show that either a knot is the unknot or that the knot is not the unknot. The $\text{NP}$ certificate uses normal surface theory:

Ian Agol, Joel Hass, William Thurston, The computational complexity of knot genus and spanning area. Trans. Amer. Math. Soc. 358 (2006), no. 9, 3821–3850. 

The $\text{co-NP}$ certificate is currently conditional on the Generalized Riemann Hypothesis, and is based on finding representations of the knot group:

Greg Kuperberg, Knottedness is in NP, modulo GRH. Adv. Math. 256 (2014), 493–506.

(There ought to be another proof of the $\text{co-NP}$ result using normal surface theory and the notion of hierarchies. Ian Agol sketched that some time ago, but the details are difficult.)
The class $\text{NP} \cap \text{co-NP}$ is relatively small, and this gives reasons to believe that detecting the unknot is in $\text{P}$.
(I would have made this a comment, but I think it's too long for that.)