Train tracks

A few posts back, I defined normal loops in the triangulation of a surface and said I would use this idea to define train tracks on a surface. The key property of normal loops is that the normal arcs form parallel families and we can encode the topology of the curve by keeping track of how many parallel arcs are in each family. Train tracks encode loops in a surface in a very similar way. A train track is a union of bands in the surface (disks parameterized as ) with disjoint interiors, but that fit together along their horizontal sides. In other words, the top and bottom edges of each band are contained in the union of the horizontal edges of other bands. A picture of this is shown below the fold.

It came from K2

At the “Mathematics of Knots 5″ conference at Waseda University, I attended a most interesting talk by Takefumi Nosaka. Nosaka’s work always gives me the impression of being robust and sophisticated, and this talk was no exception. This time he was in the process constructing new topological invariants of links as images of longitudes in of a ring.

The Bridge Spectrum

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.)

Wise’s CBMS Lecture Notes on Cube Complexes

This fall, the topology group at OSU is reading through Dani Wise’s lecture notes on cube complexes, based on his series of talks at the CBMS-NSF conference back in 2011.  Henry Wilton and Daniel Moskovich have written on this blog about Wise’s work and its role in the proof of the Virtual Haken Conjecture. This is just a quick note to say how  impressed I’ve been with the lecture notes. They start from the very beginning, include a lot of good examples and have proved to be very accessible for all of us non-experts (which includes me).

It’s just too bad that Wise’s notes are no longer available on the conference web page. (Now that a paper copy is available from the AMS, the PDF file has been replaced with a note saying that the editor insisted they be taken down.)  You can still e-mail Dani Wise to request a copy, but I expect that some people (such as beginning graduate students) might be reluctant to e-mail someone they don’t know like this. I can assure you, he was very gracious when I asked him for a copy and seems to be very eager to distribute the notes widely. But, if you have any thoughts on how the PDF file could be distributed more efficiently, I would love to hear about it in the comments.

Topologically minimal surfaces – More common than you might think

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.

Normal loops in surfaces

I plan to write a post or two about normal surfaces and branched surfaces in three-dimensional manifolds, but I want to warm up first, with two posts about the two-dimensional analogues of these objects. Train tracks play a huge role in the approach to the topology of surfaces initiated by Nielsen and Thurston, for understanding mapping class groups, Teichmuller space, laminations, etc. They organize the set of isotopy classes of simple closed curves in a surface in a way that allows one to take limits of infinite sequences of loops. (The limits are called projective measured laminations.)  In this post and the next, I will discuss train tracks from a rather unusual perspective, via normal loops in a triangulation of the given surface.

Morse-Novikov number and tunnel number

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.

Digital spheres and Reeb graphs

This post is about a slightly different direction in the topology of data sets than what I’ve written about previously, but one that is much more explicitly low-dimensional. When a doctor takes an MRI scan of a patient’s brain, they get a three dimensional lattice of points labeled foreground (brain) and background (not brain). The boundary between these two sets should be a sphere, and this is important for things like mapping regions of the brain to biological functions. However, because the MRI is not 100% accurate, even if your brain is a sphere, the MRI data may not be. Two electrical engineers (David Shattuck and Richard Leahy [1]) proposed a method to determine if the boundary of a brain scan is a sphere by building a certain graph from the cross sections of the scan. This seems to be one of the rare instances in which a cleanly stated math problem just drops out of science and engineering. Shortly after this, three mathematicians (Lowell Abrams, Donniell Fishkind and Carey Priebe [2]) picked the conjecture up, proved it, then proved a number of follow-up results. Below the fold, I’ll describe the conjecture and how it’s related to a classical topological tool – the Reeb graph of a Morse function.

SnapPy 1.7: Ptolemy and reps to PSL(n, C).

SnapPy 1.7 is out. The main new feature is the ptolemy module for studying representations into PSL(n, C). This code was contributed by Mattias Görner, and is based on the the following two very interesting papers:

1. Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert: Gluing equations for PGL(n,C)-representations of 3-manifolds.
2. Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert: The complex volume of SL(n,C)-representations of 3-manifolds.

You can get the latest version of SnapPy at the usual place.

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

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.