# Low Dimensional Topology

## November 26, 2013

### What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”.  It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

## July 11, 2013

### Smooth proof of Reidemeister-Singer

Every construction I know of 3-manifold invariants from Heegaard splittings factors through the Reidemeister-Singer Theorem:

Reidemeister-Singer Theorem: For any two Heegaard splittings $H_1$ and $H_2$ of a 3-manifold $M$, there exists a third Heegaard splitting $H$ which is a stabilization of both.

This theorem is definitely part of the big story in 3-manifold topology, and is usually proven in the PL category, as for example in Nikolai Saveliev’s Lectures on the Topology of 3-manifolds. There is another nice PL proof due to Craggs, Proc. Amer. Math. Soc. 57, n 1 (1976), 143-147.

I think of a Heegaard splitting as being intrinsically a smooth topology construction (a level set of a Morse function), and so I would really like the proof of Reidemeister-Singer to live in the smooth category. I think that there should be consistent smooth and PL stories of 3-manifold topology living side by side. In the 1970’s, Bonahon wrote a smooth proof of Reidemeister-Singer, which uses Cerf Theory (naturally, because we’re investigating paths between Morse functions). Unfortunately, Bonahon’s proof was never published, and it is lost.

A year ago (but I only saw it this morning), François Laudenbach posted a smooth proof of Reidemeister-Singer to arXiv: http://arxiv.org/abs/1202.1130. I think that this is wonderful! There are too few papers like this- there is insufficient incentive to streamline the storylines of foundations. I am very happy to have found this proof, and I want such a proof to be a part of my smooth 3-manifold topology foundations.

Edit: Thanks to George Mossessian and to Ryan Budney, who point out in the comments that Jesse Johnson proved Reidemeister-Singer using Rubinstein and Scharlemann’s sweep-outs, which involves singularity theory which is much less sophisticated that Cerf Theory: http://front.math.ucdavis.edu/0705.3712
Perhaps that should be the “smooth proof from The Book” (or the “proof from The Smooth Book”)!

## 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.

## April 3, 2013

### Update on subadditivity of tunnel number

Filed under: Heegaard splittings,Knot theory — Jesse Johnson @ 12:54 pm

A few months ago, I wrote a blog post about the interesting phenomenon that the tunnel number of a connect sum of two knots may be anywhere from one more than the sum of the tunnel numbers to a relatively small fraction of the sum of the tunnel numbers. Since then, a couple of related papers have been posted to the arXiv, so I thought that justifies another post on the subject. The first preprint I’ll discuss, by João Miguel Nogueira [1], gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer [2] (who is currently a postdoc here at OSU), gives a new bound on the amount tunnel number and Heegaard genus can degenerate by under connect sum/torus gluing, respectively, in certain situations.

## February 16, 2013

### The Bridge Spectrum

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 9:37 pm

A knot $K$ in a three-manifold $M$ is said to be in bridge position with respect to a Heegaard surface $\Sigma$ if the intersection of $K$ with each of the two handlebody components of the complement of $\Sigma$ is a collection of boundary parallel arcs, or if $K$ is contained in $\Sigma$. The bridge number of a knot $K$ in bridge position is the number of arcs in each intersection (or zero if if $K$ is contained in $\Sigma$) and the genus $g$ bridge number of $K$ is the minimum bridge number of $K$ over all bridge positions relative to genus $g$ Heegaard surfaces for $M$. The classical notion of bridge number is the genus-zero bridge number, i.e. bridge number with respect to a sphere in $S^3$, but a number of very interesting results in the last few years have examined the higher genus bridge numbers. Yo’av Rieck defined the bridge spectrum of a knot $K$ as the sequence $(b_0,b_1,b_2,\ldots)$ where $b_i$ is the genus $i$ bridge number of $K$ and asked the question: What sequences can appear as the bridge spectrum of a knot? (At least, I first heard this term from Yo’av at the AMS section meeting in Iowa City in 2011 – as far as I know, he was the first to formulate the question like this.)

## January 25, 2013

### Topologically minimal surfaces – More common than you might think

Filed under: 3-manifolds,Heegaard splittings,Thin position — Jesse Johnson @ 12:02 pm

Before I get back to train tracks (as I had promised in my last post), I wanted to point out some interesting recent work on topologically minimal surfaces. The definition of topologically minimal surfaces was introduced by Dave Bachman [1] as a topological analogue of higher index geometrically minimal surfaces, suggested by work of Hyam Rubinstein. I discussed these in detail in my series of posts on axiomatic thin position, but here’s the rough idea: An incompressible surface has topological index zero because there is no way to compress it, so it’s similar to a local minimum, i.e. an index-zero critical point of a Morse function. A strongly irreducible Heegaard surface has topological index one because there are two distinct ways to compress it, similar to how there are two distinct ways to descend from an index-one critical point (a saddle) in a Morse function. An index two surface will be weakly reducible, but there will be an essential loop of compressions, in the sense that consecutive compressing disks will be disjoint, but the loop is homotopy non-trivial in the complex of compressing disks. This should remind you of an index-two critical point in a Morse function, in which there is a loop of directions in which to descend. Then index-three surfaces have an essential sphere of compressions and so on. Initially, it was unclear how common higher index surfaces would be. I would have guessed that they weren’t very common, and I think Dave felt the same. But a number of recent results indicate quite the opposite.

## December 18, 2012

### Morse-Novikov number and tunnel number

Filed under: 3-manifolds,Heegaard splittings,Knot theory,Thin position — Jesse Johnson @ 9:33 am

Someone recently pointed out to me a paper by A. J. Pajitnov [1] proving a very interesting connection between circular Morse functions and (linear) Morse functions on knot complements. (A similar result is probably true in general three-manifolds as well.) Recall that a (linear) Morse function is a smooth function from a manifold to the line in which there are a finite number of critical points (where the gradient of the function is zero), and each critical point has one of a number of possible forms. For a two-dimensional manifold the possible forms are the familiar local minimum, saddle or local maximum. This post is about three-dimensional Morse functions, in which case the possible forms are slight generalizations of local minima, maxima and saddles.  A circular Morse function is a function with the same conditions on critical points, but whose range is the circle rather than the line. For a three-dimensional manifold, the minimal number of critical points in a linear Morse function is twice the Heegaard genus plus two, and for knot complements it’s twice the tunnel number plus two. (In particular, one can construct a Heegaard splitting or unknotting tunnel system directly from a Morse function, but that’s for another post.) The minimal number of critical points in a circular Morse function is called the Morse-Novikov number, and is equal to the minimal number of handles in a circular thin position for the manifold (usually a knot complement). Pajitnov has a very clever argument to show that the (circular) Morse-Novikov number of a knot complement is bounded above by twice its (linear) tunnel number. Below, I want to outline a slightly different formulation of this proof in terms of double sweep-outs, though I should stress that the underlying idea is the same.

## October 23, 2012

### More than you probably wanted to know about Scharlemann’s no-nesting Lemma

Filed under: 3-manifolds,Heegaard splittings,Thin position — Jesse Johnson @ 11:38 am

This post is going to be a bit more technical than usual (though not necessarily any more coherent). As I’ve been working on porting thin position techniques to the analysis of large data sets and other arenas, I’ve had to spend a lot of time trying to understand how the fundamental ideas fit together, and one in particular is Scharlemann’s no-nesting Lemma. This Lemma says the following: Given a strongly irreducible Heegaard surface $\Sigma$ and an embedded disk $D$ with essential boundary in $\Sigma$, you can always make the interior of $D$ disjoint from $\Sigma$ by isotoping away disks and annuli in $D$ that are parallel into $\Sigma$. As I’ll describe below, it turns out that this Lemma in many ways encapsulates the fundamental properties of thin position.

## August 22, 2012

### Bill Thurston is dead at age 65.

Bill Thurston passed away yesterday at 8pm, succumbing to the cancer that he had been battling for the past two years.   I don’t think it’s possible to overstate the revolutionary impact that he had on the study of geometry and topology.  Almost everything we blog about here has the imprint of his amazing mathematics.    Bill was always very generous with his ideas, and his presence in the community will be horribly missed.    Perhaps I will have something more coherent to say later, but for now here are some links to remember him by:

## November 29, 2011

### The minimal genus Heegaard splitting conjecture

Filed under: 3-manifolds,Heegaard splittings — Jesse Johnson @ 4:49 pm

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 $g$, there is a constant $K_g$ such that if $M$ is a hyperbolic 3-manifold with Heegaard genus $g$ then $M$ has at most $K_g$ isotopy classes of (minimal) genus $g$ Heegaard splittings.

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