One of my mathematical siblings, Mike Williams, recently posted step one of his three step program to prove the Berge conjecture for tunnel-number-one knots. The Berge conjecture, which was mentioned in a previous post, states that if non-trivial Dehn surgery on a knot in the 3-sphere produces a lens space then the knot should be double primitive in a genus two Heegaard splitting of the 3-sphere. (Such a knot is called a Berge knot.) All Berge knots happen to be tunnel-number-one, i.e. there is an arc with its endpoints in the knot such the the complement of the arc and the knot is a genus two handlebody. Mike wants to show that every tunnel-number-one knot with a Dehn surgery that produces a lens space is a Berge knot.
The idea behind the program is that a double primitive knot is isotopic to a core of each handlebody in a genus two Heegaard splitting of the 3-sphere. A tunnel-number-one knot is primitive in the “inside” handlebody in such a Heegaard splitting but not necessarily in the “outside” handlebody. It happens that a loop in the boundary of a genus two handlebody is primitive if and only if gluing at 2-handle along that loop produces a genus one handlebody (a solid torus – this is why Berge knots produce lens spaces). Mike’s idea is to push the tunnel number one knot into the Heegaard surface so that its slope of intersection with the surface agrees with the Dehn surgery that produces a lens space. Then look at just the outside handlebody with this loop in its boundary. One wants to show that gluing a two handle along this loop produces a solid torus.
In general, gluing along the loop in the outside handlebody produces a 3-manifold with a genus two Heegaard splitting and a single torus boundary component. This 3-manifold is either a Seifert fibered space, a toroidal manifold or a hyperbolic manifold. Mike’s current paper shows that in the case when attaching a 2-handle produces a Seifert fibered space, it must actually produce a solid torus. A second paper proving the toroidal case is in its final stages. Both proofs rely on the classification of Heegaard splittings of the manifolds involved (small Seifert fibered spaces and graph manifolds, respectively). Heegaard splittings of one-handle hyperbolic 3-manifolds are not well understood (as far as I know) so the hyperbolic case is proving the most difficult.