Low Dimensional Topology

September 29, 2011

The generalized Scharlemann-Tomova conjecture

Filed under: 3-manifolds,Curve complexes,Heegaard splittings — Jesse Johnson @ 6:18 am

Soon after John Hempel introduced the notion of (curve complex) distance for Heegaard splittings, Kevin Hartshorn showed that the existence of an incompressible surface implies a bound on the distance for any Heegaard splitting of the same 3-manifold. Scharlemann and Tomova noted that a strongly irreducible Heegaard surface behaves much like an incompressible surface, and generalized Hartshorn’s Theorem as follows: If M admits a distance d Heegaard surface \Sigma then every other genus g Heegaard surface with 2g < d is a stabilization of \Sigma. This is a great theorem and has had huge consequences for the field, but there is one thing that has always bothered me about it: It leaves open the possibility that, for example, there may a 3-manifold M with a genus three, distance 100 Heegaard surface and a second Heegaard surface of genus 201 that is unrelated to the first one. This has always seemed very unlikely to me, so I propose the following conjecture:

The generalized Scharlemann-Tomova conjecture: For every genus g, there is a constant K_g such that if \Sigma \subset M is a genus g, distance d \geq K_g Heegaard surface then every Heegaard surface for M is a stabilization of \Sigma.

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June 21, 2013

Lots and lots of Heegaard splittings

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 12:28 pm

The main problem that I’ve been thinking about since graduate school (so around a decade now) is the following: How does the topology of a three-dimensional manifold determine its isotopy classes of Heegaard splittings? Up until about a year ago, I would have predicted that most three-manifolds probably don’t have many distinct Heegaard splittings, maybe even just a single minimal genus Heegaard splitting and then all of its stabilizations. Sure, plenty of examples have been constructed of three-manifolds with multiple distinct (unstabilized) splittings, but these all seemed a bit contrived, like they should be the exceptions rather than the rule. I even wrote a blog post a couple years back stating what I called the generalized Scharlamenn-Tomova conjecture, which would imply that a “generic” three-manifold has only one unstabilized splitting. However, since writing this post, my view has changed. Partially, this was the result of discovering a class of examples that disprove this conjecture. (I’m hoping to post a preprint about this on the arXiv in the near future.) But it turns out there is an even simpler class of examples in which there appear to be lots and lots of distinct Heegaard splitting. I can’t quite prove that they’re distinct, so in this post I’m going to replace my generalized Scharlemann-Tomova conjecture with a conjecture in quite the opposite direction, which I will describe below.

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May 5, 2011

Why cut disks?

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 1:26 pm

Classical thin position for links is a relatively simple idea:  Draw a link diagram on a piece of paper, then think of a horizontal red line sliding from above the kink to below, keeping track of the number of points of intersection at each time that it’s transverse to the link.  To get to thin position, we want to redraw the diagram so that the number of intersections at any given point stays as small as possible.  This perspective makes it seem like thin position is about isotoping knots around, but Scharlemann and Thompson noticed that it’s actually about surfaces: If you think of those red lines as the projections of horizontal spheres, then thin position is the thinnest way to push a sphere from above the knot to below it. Each time the sphere passes through the knot is defined by a bridge disk whose boundary consists of an arc in the knot and an arc in the sphere.  If you get rid of the knot and replace those bridge disks with compressing disks, you get Scharlemann and Thompson’s thin position for 3-manifolds [1].

Hayashi and Shimokawa [2] took this a step further by defining a type of thin position in which one looks at surfaces in an arbitrary 3-manifold containing a link and considers both compressing disks and bridge disks.  (The relationship between these different types of thin positions is the basis for the axiomatic thin position that I wrote about earlier, and which is now carefully written up in a preprint.)  Maggy Tomova has generalized Hayashi-Shimokawa thin position even more (and used this type of thin position very effectively) by adding into the mix what she calls a cut disk – a disk whose boundary is in the surface and whose interior intersects the link in a single point.  It has taken me a long time to get used to and appreciate this idea, but now that I realize how perfectly cut disks fit in with the general thin position machinery, I would like to try and explain it below.

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March 10, 2009

Thin position and graphs

Filed under: 3-manifolds,Heegaard splittings,Triangulations — Jesse Johnson @ 2:51 pm

Today I want to write about an application of axiomatic/iterated thin position that connects thin position very strongly to the theory of normal/almost normal surfaces.  This is the application that actually got me started thinking about thin position, and which I’m writing up in paper-form as I write these posts.    This will probably be my last post about thin position for a while, so before beginning my next post on axiomatic thin position, here’s a quick synopsis of what I’ve covered so far:

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