I mentioned in a previous post recent work by Cho and McCullough about the tree of unknotting tunnels. At the AIM a few weeks ago, Martin Scharlemann discussed their work and some of the earlier work that led to it, in particular his paper with Goda and Thompson [1]. They showed that given a tunnel-number-one knot in minimal bridge position, one can always slide the tunnel around so that it sits in a level sphere. While sliding the tunnel around, one may need to pass the end points of the tunnel past each other along the knot. If we think of the theta graph (the knot union its tunnel) as the spine of a genus two handlebody, sliding the endpoints past each other corresponds to picking new meridians for the handlebody. (The meridian dual to the tunnel is uniquely determined, but the other two meridians are not.) By making the tunnel level, we choose a pair of meridians for the knot relative to the tunnel.

In Cho and McCullough’s work [2], one finds a sequence of theta graphs, starting with a standardly embedded graph, related by simple moves such that the final theta graph is isotopic to the knot union tunnel that you were looking for. Their theorem states that this sequence (the way they define it) is unique. In his talk at the AIM, Scharlemann pointed out that because the sequence is unique, the final pair of meridians for the knot relative to the tunnel are uniquely determined. Moreover, it turns out that this pair of meridians is the same as the pair that one gets by leveling the tunnel a la Goda-Scharlemann-Thompson.

In my previous post about Cho and McCullough’s result, I mentioned the (poorly defined) problem of trying to understand how the cabling moves they define affect the knot complement. David Futer has suggested one way to interpret the moves might be to put a cone hyperbolic metric on the knot complement. Then when one carries out the cabling move, one deforms the metric around the tunnel into a cone, then continues deforming until another edge of the theta graph becomes non-cone points. This might also be useful in answering the question of whether every unknotting tunnel in a hyperbolic tunnel-number-one knot is isotopic to a geodesic. More immediately, though, it suggests the following question: If we have a tunnel in a hyperbolic knot that happens to be a geodesic in the knot complement, does it define a unique pair of meridians for the knot relative to the tunnel, and if so are these the same as those defined by leveling the tunnel with respect to a bridge sphere?

The question of whether the geodesic determines a unique pair of meridians really has to do with how the hyperbolic manifold is embedded in the knot complement. It shouldn’t be too hard to work out, but I’ll leave that for the geometers to work out. The second part of the question above is probably much harder.

[1] H. Goda, M. Scharlemann, and A. Thompson. Leveling an unknotting tunnel. Geometry and Topology, 4:243–275, 2000. ArXiv:math.GT/9910099.

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