Today, I will continue on my quest to find the most interesting conjectures about Heegaard splittings. (Most of these conjectures, including this one, fail criteria one and two in Daniel’s recent post, but strive to satisfy criteria three.) Here’s the latest:
The minimal genus Heegaard splitting conjecture: For every positive integer , there is a constant such that if is a hyperbolic 3-manifold with Heegaard genus then has at most isotopy classes of (minimal) genus Heegaard splittings.
Note that the hyperbolic condition is necessary because there are toroidal manifolds, particularly Seifert fibered, spaces with infinitely many minimal genus Heegaard splittings . Minimal genus (rather than just irreducible) is also necessary since there are hyperbolic manifolds with infinitely many distinct irreducible splittings . On the other hand, Lustig and Moriah have constructed manifolds with arbitrarily many minimal genus Heegaard splittings , but in order to increase the number of splittings, they have to increase the genus of the the Heegaard splittings. Note that Lustig and Moriah’s examples show that if exists, it grows at least exponentially with . I’ve been toying lately with constructions that produce many distinct irreducible splittings, but again I need to increase the genus in order to increase the number of minimal genus splittings.
My intuition is that for a relatively simple hyperbolic 3-manifold, there are relatively few ways to (efficiently) cut up its topology. In order to find a manifold that can be cut up in more ways, you need to make it more complicated, which increases its genus. However, “more complicated” in this case can’t mean simply higher volume because there are hyperbolic 3-manifolds of arbitrarily high volume with bounded Heegaard genus, though their minimal genus Heegaard splittings will, generally speaking, be unique.
Instead, the picture of what a 3-manifold with lots of minimal genus Heegaard splittings should look like seems to be related to the conjecture that Ian Biringer wrote about a few years ago here. The conjecture is, roughly, that all hyperbolic manifolds with bounded rank and injectivity radius are made up of a finitely many types of pieces glued together along their (incompressible) boundaries. More “complcated” gluing maps should correspond to higher volume manifolds, but once you choose the types of pieces and pair up their boundary components, there will be a bound on the number of minimal genus Heegaard splittings. To increase this number, you would need to either choose more blocks (which increases the genus) or choose more complicated blocks by allowing higher rank (and higher genus) or lower injectivity radius.
 Unpublished preprint by Andrew Casson and Cameron Gordon. See this paper for a generalization of the same construction.
 Lustig, Martin and Moriah, Yoav. On the complexity of the Heegaard structure of hyperbolic 3-manifolds. Math. Z. 226 (1997), no. 3, 349–358.