Unknotting algorithm in higher dimensions?

Question

Suppose we are given a 2-knot (say by a movie). Is there an algorithm to tell if it is unknotted ? I suppose that it could matter if I say "topologically" or "smoothly" here since those could be different - I am interested in results in either direction.

Is there an algorithm to tell if the fundamental group of the complement is $\mathbb{Z}$?

While I am mainly interested in the 4-dimensional case, I imagine the above problems are hard (although I can't find a reference) - maybe something is known in higher dimensions? Here there is an algebraic characterization (due to Kervaire) of the knot complements that can occur (namely finitely-presentable groups, generated by a single conjugacy class, has cyclic first homology, and 0 second homology), so maybe there is a result that says that in this class of groups there is no algorithm to recognize $\mathbb{Z}$?

Answer 1

The version of the problem in dimensions higher than 4 is undecidable, by work of Nabutovsky and Weinberger: https://link.springer.com/content/pdf/10.1007/BF02566428.pdf
This is related to the undecidability of the triviality problem for group presentations.

The 4-dimensional version is thought to be undecidable, but this is not known. The relevant question in geometric group theory is: is it decidable whether a "balanced" group presentation (one that has the same number of generators and relations) presents the trivial group? Or $\mathbb{Z}$? For some limited recent progress on this problem, see this paper by Lishak and Nabutovsky: https://arxiv.org/abs/1510.02773

This is all in the smooth or PL case. For topological knots, it's not clear to me how you would represent them computationally in a finite way. (Perhaps there are even uncountably many types of them? Someone who is better with topological topology should chime in here.)