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](http://ams.org/mathscinet/search/publdoc.html?extend=1&pg1=IID&s1=203701&vfpref=html&r=16&mx-pid=1628753) (and in [this paper](http://ams.org/mathscinet/search/publdoc.html?extend=1&pg1=IID&s1=203701&vfpref=html&r=17&mx-pid=1664979)), 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?



Unknotting tunnels in surface bundles
[Scharlemann and Thompson](http://www.ams.org/mathscinet-getitem?mr=1990938) proved that one can isotope/slide a tunnel
to be disjoint from a minimal genus Seifert surface (which was what I had in mind
in my comment above). Their argument does not depend on it being a knot complement.
I also found a [citation for a paper of Sakuma](http://www.ams.org/mathscinet-getitem?mr=1483404) which claims to prove the
result for punctured torus bundles, so it might have the result you're looking
for (although I haven't looked at the article yet). 


