I want to mention the upcoming special session at the Joint Mathematics Meetings in San Diego for which I’m one of the organizers. (Abby Thompson and Robin Wilson are the other two.) We’ve got a whole day of talks ranging from old pros like John Hempel and Hyam Rubinstein to grad students just entering the field. (See the official program.) The title of the session, “Heegaard splittings, bridge positions and low dimensional topology”, refers to the fraternal relationship between Heegaard splittings of 3-manifolds and bridge positions (or bridge surfaces) for knots and links. A bridge surface for a link (which doesn’t appear to have a wikipedia entry) is a Heegaard surface for the ambient 3-manifold such that the link intersects each handlebody of the Heegaard splitting in boundary parallel arcs. (This definition is a generalization of the classical definition, in which the Heegaard surface is a sphere in S^3.)
The connection between Heegaard surfaces and bridge surfaces is initially motivated by the fact that in a branched cover over a link, a bridge surface for the link lifts to a Heegaard surface for the cover. In a few simple cases, this yields a one-to-one correspondence, but in most cases things aren’t so nice. (It breaks down because one has to consider equivariance.) There is still a strong analogy between the two setting, first demonstrated by Gabai’s generalization of bridge position to thin position for links. Scharlemann and Thompson showed that translating this definition to thin position for Heegaard splittings led to a more intuitive proof of an important theorem of Casson and Gordon. Thompson later translated Casson and Gordon’s theorem into the framework of bridge surfaces.
Since then, there have been a number of theorems that have been translated from one setting to the other. The goal of the special session is to find more ways in which our knowledge of one of the settings can be used to improve our knowledge of the other. The session runs a full day, starting with talks about knots and link in the morning, then a couple of talks about stabilizations of Heegaard splittings right before lunch. After lunch, we have talks about the curve complex, beginning with their applications to Heegaard splittings and then with their applications to tunnel-number-one knots. Following that, it’s two talks about surface automorphisms and handlebodies/Heegaard splittings, then a couple of miscellaneous talks (degree one mappings and hyperbolic geometry, respectively) and then two talks related to the Berge conjecture.