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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?

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    $\begingroup$ I think this result should generalize to fibered manifolds of higher genus, using methods of Jesse Johnson (who obtained a classification in the closed case): front.math.ucdavis.edu/0607.5513 Alternatively, one may analyze how a compressing disk crosses over the tunnel, and use it to simplify the intersections of the tunnel with the fiber so that it is disjoint. Then an outermost disk gives an isotopy of the tunnel onto the fiber. Another outermost disk shows one may find a disjoint representative on the fiber after applying $\phi$. $\endgroup$
    – Ian Agol
    Commented Oct 4, 2010 at 16:13
  • $\begingroup$ The link in Ian's comment is broken, here's a replacement: arxiv.org/abs/math/0607513 $\endgroup$
    – David Roberts ♦
    Commented Mar 29, 2022 at 7:30

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Scharlemann and Thompson 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 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).

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The classification of once-punctured torus bundles over $S^1$ with Heegaard genus two should follow directly from a paper of Cooper and Scharlemann. Jesse's paper (above) uses similar techniques but deals with general genus fiber (harder!). Thinking about Ian's outermost disk suggestions is probably a good idea.

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