After thinking some more about my post yesterday on Heegaard splittings of amalgamated 3-manifolds, a slightly more general (and much more arcane) conjecture came to mind. I discussed the phenomenon that given the separating F surface where the two submanifolds are glued, there are some Heegaard surfaces that contain subsurfaces parallel to F minus sum punctures. These are amalgamations of Heegaard splittings for the complementary components and their existence is unrelated to the gluing map between the two submanifolds. There are also Heegaard surfaces that do not have subsurfaces parallel to F (They’re not amalgamations along F.) and the existence of such Heegaard surfaces depends entirely on the gluing map. In particular the genera of such surfaces are bounded below by a certain distance in the curve complex associated to the gluing map.
I didn’t mention this last time, but the main (in some sense the only) technique for finding these high distance gluing maps is to compose a given gluing map with a high power of a pseudo-Anosov map. Let’s say that instead, we were to compose the gluing map with a reducible automorphism of F that is pseudo-Anosov on a subsurface G of F and the identity everywhere else. We get a sequence M_i of 3-manifolds, each containing a copy of F. The distances of these embedded surfaces are bounded, so it’s conceivable that every M_i has a Heegaard surface of bounded genus that is not an amalgamation along F.
In fact, if we can find a Heegaard surface S for M such that a subsurface of S coincides with the subsurface G of F then the image of S in each M_i will be a Heegaard surface. I would like to conjecture that this is the only way one can get bounded genus Heegaard splittings for the M_i. In other words, for each genus g, there should be some K such that for k > K, every Heegaard surface S for M_k of genus greater than g can be isotoped so that a subsurface of S coincides with G. In the case when G = F, this is precisely Li’s result that I discussed last time.