What is the complexity of determining if a knot group is $\mathbb{Z}$?

Question

It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:

1. The complexity of determining if a knot group is $\mathbb{Z}$? .i.e. same as the unknot.
2. The complexity of the isomorphism algorithm in general.

Regards, Prathamesh

Answer 1

The knot group is $\mathbb{Z}$ if and only if the knot is unknotted: see https://math.stackexchange.com/questions/1478171/knot-group-and-the-unknot

So, your first question is on the complexity of deciding whether a knot is the unknot. This is known to be in the intersection of NP and co-NP, see the Wikipedia article on this any many other interesting facts.