The computational complexity of the solid torus core recognition problem

Abstract

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 $\mathbf{\text{NP}}$ since the thickened torus recognition problem is in $\mathbf{\text{NP}}$. We give an alternate proof of the fact and prove that the problem is in $\mathbf{\text{co-NP}}$. It is also proved that the Hopf link recognition problem is in $\mathbf{\text{NP}}$ and $\mathbf{\text{co-NP}}$ as a corollary to our result.