Low Dimensional Topology

October 26, 2009

Genus two Heegaard splittings

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 12:32 pm

I’ve been distracted away from blogging in the last few months, but there have been some recent additions to the arXiv that I couldn’t resist writing about.  The most recent reconsiders a paper by Rubinstein and Scharlemann [1] about genus two Heegaard splittings.  Rubinstein and Scharlemann applied their double sweep-out/graphic method to show that two distinct Heegaard splittings of the same 3-manifold can be made to intersect in a relatively simple manner, which allows them to characterize how the two Heegaard splittings are related.  In particular, the hyper-elliptic involutions on the 3-manifold determined by the two Heegaard surfaces  commute.

However, it appears that their analysis missed a case.  The gap in their proof and a class of examples that fall outside the original classification were discovered by John Berge, using double-primitive knots.  The present preprint [2] by Berge and Scharlemann presents these examples and repairs the proof.  There are two interesting properties of the new examples.  First, some of them have distance three (the originals only had distance two) and second, it is not clear that the new pairs become isotopic after a single stabilization (it was for the originals).  For all the examples, new and old, it is unknown when the pair of Heegaard surfaces are actually non-isotopic.  (They only prove that if a pair of genus two splittings are not isotopic then they fall into one of these cases.)  If one could show that the distance three Heegaard splittings are in fact distinct, this would be the first example of a distance three Heegaard splitting that is not unique.  (All the examples of distinct Heegaard splittings where the distance is known have distance two, though there are other examples where the distance has not been calculated.)  If one could show that the new examples do, in fact, require more than one stabilization then this would be the first example where the stable genus of two Heegaard splittings is equal to the sum of their genera.  Either of these results would be very interesting.

Two more preprints that caught my attention are a pair of papers by Scott Taylor and Maggy Tomova [3],[4] that generalize Hayashi-Shimokawa thin position and Tomova’s c-disk thin position to graphs.  They use this to consider the problem of leveling an edge of a graph into a Heegaard surface.  This is related to a paper by Goda, Scharlemann and Thompson showing that given a tunnel-number-one knot in minimal bridge position in the 3-sphere, the unknotting tunnel can be isotoped while fixing the knot, so that the tunnel is contained in the bridge sphere [5].  (I wrote about this previously.)  Cho and McCullough, for example, use this result to bound the bridge number of a knot in terms of its position in the tree of unknotting tunnels [6].  The proof by Goda, Scharlemann and Thompson uses the classical version of thin position in which every level surface is a single sphere.  The more recently developed forms of thin position, in which spheres can be compressed and cut-compressed, have proven much more natural, as seen for example in [7].  So I have high hopes that this new approach will prove useful for applications such as, for example, leveling an unknotting tunnel with respect to a higher genus Heegaard surface for the 3-sphere.

Finally, I wanted to point out that Brandy Guntel has posted a preprint [8] with examples of knots that have distinct double-primitive and primitive/Seifert positions with the same surface slope in a genus two Heegaard splitting of the 3-sphere.  (I wrote about some of her examples previously.)  The knots are all twisted torus knots and the (computationally intense) argument is based on calculating the action of the Goeritz group on the homology of the Heegaard surface.

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