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 $\mathbb{Z}$?