I’ve been meaning to write about a recent preprint of Kobayashi and Rieck [1] that improves a result of Saul Schleimer’s [2]. Saul showed that for every 3-manifold, there is a value k such that every Heegaard splitting for that manifold of genus greater than k has the disjoint curve property (i.e. Hempel distance at most 2). Schleimer’s bound is an exponential function in terms of the number of tetrahedra in a minimal triangulation for the 3-manifold. Kobayashi and Rieck have improved the bound to a linear function of the number of tetrahedra.

These results are most interesting in the context of Tao Li’s work [3] [4] on branched surfaces and Heegaard splittings. Li showed (roughly) that every 3-manifold has a finite family of Heegaard splittings such that every irreducible Heegaard splitting for the manifold comes from Haken summing a surface from this finite family with a collection of incompressible surfaces. Combinging this with Schleimer’s result implies that in an atoroidal 3-manifold there are finitely many high distance splittings. The splittings that come from repeated Haken summing (and therefore have high genus) must all have the disjoint curve property. This suggests that there is a sort of fundamental distinction between the finitely many high distance splittings and the possibly infinitley many low distance ones.

Both Schleimer’s proof and Kobayahsi-Rieck’s proof use normal surface theory. Recall that a surface in a 3-manifold is normal with respect to a given triangulation if it intersects each tetrahedron in a collection of (normal) triangles and quadrelaterals. A surface is almost normal if it intersects each tetrahedron in a collection of triangles and quadrelaterals plus its intersection with exactly one tetrahedron also contains an octagon or an annulus whose boundary loops each intersect three or four edges.

Notice that there are two types of almost normal surfaces: those with octagon pieces and those with annulus pieces. If you will allow me to descend into sheer speculation, I’d like to suggest that there should be some sort of connection between the octagon/annulus dichotomy and the high distance/low distance dichotomy.

This speculation is motivated by the fact that in a tube almost normal surface, compressing along the tube produces a normal surface that bounds a handlebody on one side. If this handlebody is not a regular neighborhood of a subcomplex of the triangulation then the normal structure on the surface induces a two dimensional spine for the handlebody that has no order one edges. This two dimensional complex is homtotopy equivalent to a graph, but it is not collapsible so it’s a higher genus version of something like the house with two rooms. In a reasonable triangulation, one would hope to be able to avoid this sort of pathological behavior. (Note that for an octagon normal surface, the induced spines on both handlebodies have order one edges, so they could easily be collapsible onto graphs.)

I don’t know enough about normal surfaces to suggest a specific conjecture or question that would sum up what I’m trying to get at. I think it’s very unlikely that every octagon normal surface has the disjoint curve property and I know that every tube almost normal surface will not have high distance. I will thus leave it as a vague suggestion that there should be a more subtle connection lurking just in the background of all this normal surface/Hempel distance business.

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