The title is just about it. Assume we have a nontrivial knot $K$ in $S^3$ and the exterior of $K$, $E(K)$, is $S^3 \setminus N(K)$. Here $N(K)$ is a regular neighborhood.

- Let $\tau$ be a properly embedded arc in $E(K)$ and let $M = E(K)\setminus N(\tau)$. Now, if we know that $\pi_1(M) = \langle x,y\vert \rangle$, does this necessarily mean that $\tau$ is an unknotting tunnel and the tunnel number of $K$ is one, $t(K)=1$?
- I would like to know this in a more general setting, but the simplest case is all I really need for now. Given $T = \{\tau_1,\ldots,\tau_j\}$ properly embedded disjoint arcs, and $M = E(K)\setminus N(T)$ with $\pi_1(M) $ free, does that mean that $t(K) \leq j$?

I am a little confused by the statements of Scharlemann and Thompson's Theorem 7.5 (first page of the pdf) about embedded graphs, but think that they are working in a more restrictive setting. But I could not find the answer to my question written down anywhere.