How to know if a group is the fundamental group of a knot complement and if so how would you reconstruct the original knot from it?

The following question finds the knot from a given fundamental group.

Detect a knot from its fundamental group

However my question is oriented into generally verifying if any given group is the fundamental group of a knot. And if it is what would be the process to reconstruct the knot from its given fundamental group.

By the Adian-Rabin theorem, there is no algorithm to find out if a finitely presented group is a knot group: being a knot group is an abstract property, there are knot groups (say, $\mathbb Z$), and there are finitely presented groups which cannot embed into a knot group (see, for example, Prop 1.1 here).