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 , gives new examples of knots in which the tunnel number degenerates by a large amount. The second paper, by Trent Schirmer  (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.
April 3, 2013
February 16, 2013
A knot in a three-manifold is said to be in bridge position with respect to a Heegaard surface if the intersection of with each of the two handlebody components of the complement of is a collection of boundary parallel arcs, or if is contained in . The bridge number of a knot in bridge position is the number of arcs in each intersection (or zero if if is contained in ) and the genus bridge number of is the minimum bridge number of over all bridge positions relative to genus Heegaard surfaces for . The classical notion of bridge number is the genus-zero bridge number, i.e. bridge number with respect to a sphere in , 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 as the sequence where is the genus bridge number of 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
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  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
Someone recently pointed out to me a paper by A. J. Pajitnov  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
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 and an embedded disk with essential boundary in , you can always make the interior of disjoint from by isotoping away disks and annuli in that are parallel into . 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 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
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.
October 16, 2011
Recall that the mapping class group of a Heegaard splitting is the group of automorphisms of the ambient 3-manifold that take the Heegaard surface onto itself, modulo isotopies of that keep on itself. The isotopy subgroup is the group of such maps that are isotopy trivial on , when you ignore the Heegaard surface. Hyam Rubinstein and I constructed a number of examples of Heegaard splittings with non-trivial isotopy subgroups , but all of these groups were generated by reducible automorphisms. Before our paper, Darren Long had constructed a strongly irreducible Heegaard splitting with a pseudo-Anosov element in its mapping class group  but the ambient manifold is a Seifert fibered space over a sphere with five singular fibers and Heegaard splittings of Seifert fibered spaces have large mapping class groups, usually including lots of reducible maps. This raises the question: Is there a Heegaard splitting with a non-trivial isotopy subgroup consisting entirely of pseudo-Anosov maps (other than the identity)? I will go a step further and, in the spirit of my last two posts, suggest a reckless conjecture:
The reducible automorphism conjecture: The isotopy subgroup of every Heegaard splitting is generated by reducible automorphisms.
October 5, 2011
A common problem in low-dimensional topology is to ask how the topology and geometry of a manifold changes if you glue a solid torus into one of its torus boundary components (also known as Dehn filling) or more generally, if you glue a handlebody into a higher genus boundary component. One topological version of this problem is to ask how the isotopy classes of Heegaard surfaces change. Every Heegaard surface for the unfilled manifold becomes a Heegaard surface for the filled manifold, but there may be other properly embedded non-Heegaard surfaces that also become Heegaard surfaces if you cap them off after the filling. In particular these new Heegaard surfaces may have lower genus, so the Heegaard genus of the manifold could drop after filling. The quintessential example of this is a knot complement in the 3-sphere: There are knot complements with arbitrarily high Heegaard genus, but if you Dehn fill to produce the 3-sphere, then the genus drops to zero.
Of course, for such a manifold there is exactly one filling that produces the 3-sphere and one can ask how much the genus can drop for the other fillings. There are examples where Heegaard genus drops by one for a line of slopes, and the resulting Heegaard surfaces are often called horizontal. However, Moriah-Rubinstein  (and later Rieck-Sedgwick ) showed that there are only finitely many slopes for which the genus can drop by more than one (and only finitely many lines of slopes where it drops by one.) As far as I know there are no examples where there are two slopes for which the genus drops by more than one. So one can ask:
Question: Is there a 3-manifold with Heegaard genus , a torus boundary component and two slopes on such that Dehn filling along each slope produces a 3-manifold with Heegaard genus less than or equal to ?
September 29, 2011
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 admits a distance Heegaard surface then every other genus Heegaard surface with is a stabilization of . 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 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 , there is a constant such that if is a genus , distance Heegaard surface then every Heegaard surface for is a stabilization of .