First off, if you haven’t been reading Ken Baker’s Sketches of Topology blog lately, you should take a look. In particular he just posted some great animations illustrating an open book decomposition of the three-torus from different angles.
Second, I wanted to point out a short paper that Ozsvath, Stipicz and Szabo recently posted on the arXiv . (It seems that very few of Ozsvath and Szabo’s papers are short.) Heegaard Floer theory, the other branch of mathematics with “Heegaard” in the title, is the study of a class of 3-manifold invariants that are constructed by applying advanced homological algebra techniques to Heegaard diagrams. Right now there isn’t much interplay between the folks studying Heegaard splittings and those studying Heegaard Floer homology, but not from lack of trying. In my experience, whenever mathematicians from the two camps meet, they have the same conversations trying to figure out what we can learn from each other. But it seems to come down to the fact that Heegaard Floer theory is all about looking at aspects of the 3-manifold that don’t depend on the Heegaard splitting that you started with. Those of us who just study Heegaard splittings are interested in what is different about different Heegaard splittings.
The paper above, however, is (from my point of view) a step towards bringing the two field together. It describes a construction that produces a Heegaard diagram for a given 3-manifold that is simple enough that the floer homology can be calculated by a purely combinatorial calculation. (In general, claculating Floer homology involves a “counting holomorphic disks” step that is intractable in most situations.)
The idea is to start with a presentation for the 3-manifold as a triple branched cover over a link in the 3-sphere. One then puts the link in grid position with respect to an unknotted torus T, which cuts the sphere into two solid tori. A link L is in grid position with respect to T if it is in bridge position, i.e. it intersects each solid torus in a collection of boundary parallel arcs, and the projections of these arcs into the torus form a certain nice pattern: If we take n parallel meridians on each side of the torus, their boundaries form a grid on the torus. We require that in each horizontal band of this grid, there is a single arc parallel to a bridge arc, and in each vertical band there is a single arc parallel to a bridge arc on the other side.
The meridians in the torus intersect each other in simple pattern: they cut the torus into squares. These disks are also disjoint from the link because the bridge arcs are parallel to the meridian disks. In the branched cover, the torus lifts to a Heegaard surface (because the branch set is in bridge position) and the meridian disks lift to meridian disks on opposite sides of the Heegaard surface. The squares in the torus that are disjoint from the link lift to squares in the Heegaard surface. The squares where the link passes through the torus lift to more complicated regions, but it turns out that the lifts are simple enough that the authors can modify them to get the properties that they want.
Of course, the Heegaard splittings constructed in this way will tend to have higher genus than the minimal splitting for the same 3-manifold. So there might be some interesting things to think about along the lines of comparing the Heegaard genus of a 3-manifold to the genus of the smallest splitting that comes from this construction, or the smallest splitting that has a diagram that is compatible with combinatorial Floer homology. In fact, putting a link in grid position should probably require, in many cases, more bridges than just putting it in bridge position. Perhaps I’ll add that to the open problems list since I’ve been neglecting it lately: Find knots in the 3-sphere such that the ratios between the grid number and the genus-one bridge number are arbitrarily high. (Incidentally, grid position is also a part of the Baker-Grigsby-Hedden program to prove the Berge conjecture, which I discussed previously.)