The computational complexity of the solid torus core recognition problem

The solid torus core recognition problem is the problem that, given a knot in the solid tours, decides whether the knot is the core of the solid torus. This problem is in NP since the thickened torus recognition problem is in NP. We give an alternate proof of the fact and prove that the problem is in co-NP. It is also proved that the Hopf link recognition problem is in NP and co-NP as a corollary to our result.