TITLE: What is the complexity of determining if a knot group is $\mathbb{Z}$?
QUESTION [7 upvotes]: It is known from the work of Waldhausen that the isomorphism problem for knot groups is decidable. What is then:

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

Regards,
Prathamesh

REPLY [8 votes]: 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.