Let $K \subset S^3$ be a knot. Given $r \in \mathbb{Q} \cup \{ \infty\}$, denote by $S^3_{r}(K)$ the 3-manifold obtained by Dehn surgery on $K$ with coefficient $r$. Is it true that: $S^3_r(K)$ contains a incompressible torus if and only if $K$ is a satellite with non trivial companion?
Recall that Thurston proved that a knot is a satellite with non trivial companion if and only if an essential torus is contained in its complement.