(Reprinted from my old ldt blog)

There are two gaping holes in our understanding of Heegaard splittings that, from a moral standpoint, really shouldn’t be there: First, there are no examples of closed hyperbolic 3-manifolds for which the minimal genus Heegaard splittings (let alone all Heegaard splittings) are classified. Second, there are no examples of closed three manifolds that are known to have irreducible, weakly reducible Heegaard splittings that are not minimal genus.

Moriah and Sedgwick’s recent paper [1] on twisted torus knots in the 3-sphere takes the first step towards solving the second of these problems, and perhaps indirectly towards the first one. A twisted torus knot is the image of a torus knot after integral Dehn surgery along an unknot that goes around some number of parallel strands. In other words, start with a torus knot, grab a few consecutive strands, then twist these around each other. Moriah and Sedgwick show that if the number of twists is high enough, the complement of the knot admits a unique isotopy class of genus two Heegaard splittings. (The Heegaard splitting is constructed from the one Heegaard splitting of the torus knot that is preserved by the construction that turns it into a twisted torus knot.)

The proof uses a nice result that was originally proved by Moriah and Rubinstein [2], using geometric methods, then proved again by Reick and Sedgwick [3] using purely combinatorial methods. The result says that with the exception of a certain set of slopes, every bounded genus Heegaard splitting for the result of Dehn surgery on a link is the image of a Heegaard splitting for the link complement. In the case that Moriah and Sedgwick consider, the link is the union of the torus knot and the unknot. The genus two Heegaard splittings of the link complement can be classified fairly easily. Moriah and Rubinstein’s Theorem then implies that for most Dehn surgeries on the unknot component, the only genus two Heegaard splitting will be the image of the unique genus two Heegaard splitting of the link complement.

Now, I know what your saying: What does this have to do with finding an irreducible, weakly reducible, non-minimal genus Heegaard splitting? Well, given a Heegaard splitting for a knot complement, there are two ways to get a higher genus Heegaard splitting: One is to stabilize your original splitting, but this always produces a reducible Heegaard splitting. The second way is to take a boundary parallel torus and tube it to the original Heegaard surface by a vertical arc, producing a new Heegaard surface. In many cases, this Heegaard splitting is reducible, but Moriah and Sedgwick conjecture that it may not always be. In particular, in the case they look at if this new Heegaard splitting is reducible then it is a stabilization of the original (unique) genus two Heegaard splitting. Thus to show that it is irreducible, one would only need to show that it is not isotopic to a stabilization of the genus two Heegaard splitting. It’s not obvious that this is significantly easier, but it does seem like a reasonable conjecture that this genus three Heegaard splitting might be irreducible. And a conjectured example is better than no example at all.

[1] Follow link to arXiv.

[2] Y. Moriah, H. Rubinstein, Heegaard structure of negatively curved 3-manifolds, Comm. in Ann. and Geom. 5 (1997), 375 – 412.

[3] Y. Reick , E. Sedgwick, Persistence of Heegaard structures under Dehn filling, Topology and its application 109 (2001), 41 – 53.

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