While I was in grad. school, I spent a number of months trying to generalize Goda, Scharlemann and Thompson’s result about leveling unknotting tunnels [1] (see a previous post as well) to toroidal bridge positions. Their result says that given a tunnel-number-one knot in minimal bridge position with respect to a sphere, one can isotope any unknotting tunnel (possibly sliding the endpoints past each other) into a level sphere, without increasing the bridge number of the knot. (Since the unknotting tunnel is just an arc, it can always by isotoped into a given surface. However, since its endpoints are attached to the knot, the isotopy may make the knot more complicated with respect to the surface.)
One can also consider bridge positions with respect to an unknotted torus in the 3-sphere and ask whether an unknottting tunnel can be isotoped into a level torus, again without increasing the toroidal bridge number. I started thinking about this as a way to bound the bridge number in terms of the distance in the curve complex defined by a meridian of the unknotting tunnel. It turns out there is an easier way to do this using work of Tomova. (See Moriah-Minsky-Schleimer and J.-Thompson.) However, the question of leveling an unknotting tunnel into a level torus remains open. A positive answer should, for example, lead to a classification of unknotting tunnels of (1,1) knots (knots that are one bridge with respect to an unknotted torus) and perhaps other applications.
The most obvious approach to the problem is to look at a compressing disk for the complement of the knot union tunnel (by definition this is a handlebody) and look for disjoint upper and lower disks that determine a way to reduce the bridge number of the tunnel. This is roughly the idea in Goda, Scharlemann and Thompson’s proof, though they have a lot more machinery for dealing with bridge/thin position. (In particular, they use the fact that thin position is bridge position for tunnel-number-one knots.) I can tell you from experience that this approach gets bogged down very quickly for toroidal bridge position, though perhaps with some clever new ideas it could work.
[ 1] H. Goda, M. Scharlemann, and A. Thompson. Leveling an unknotting tunnel. Geometry and Topology, 4:243–275, 2000. ArXiv:math.GT/9910099.
