Given a once-punctured surface $F$ and an orientation preserving self homeomorphism $\phi$, let $M_\phi$ be the bundle over $S^1$ with fiber $F$ and monodromy $\phi$.
In Sakuma's survey article The topology, geometry and algebra of unknotting tunnels (and in this paper), Johannson and Kobayashi are credited for proving that any unknotting tunnel for $M_\phi$ is isotopic to a tunnel $\alpha$ which lies on a fiber $F$ such that $\alpha\cap\phi(\alpha)=\emptyset$. This leads to a classification of unknotting tunnels in once-punctured torus bundles.
The references are talks. Does anyone know if this is written down anywhere?