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.
January 25, 2013
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.
July 25, 2011
A few months ago, I wrote about a generalization of thin position for bridge surfaces of knots and links (introduced by Maggy Tomova ) that comes from considering cut disks, i.e. compression disks that intersect the link in a single point in their interior. One implication of this new definition is that it suggests a generalized concept of stabilization. A stabilization of a Heegaard surface is defined by a pair of compressing disks on opposite sides of the Heegaard surface whose boundaries intersect in a single point. Compressing along either of these disks produces a new Heegaard surface with lower genus. A cut stabilization occurs when one of those disks is a cut disks, and this situation comes up, for example, in Taylor and Tomova’s version of the Casson-Gordon Theorem . As I will describe below the fold, cut compressions have very interesting implications for studying bridge surfaces of different genera.
May 30, 2011
At the AMS Sectional in Iowa City a few months ago, there were a number of talks about circular thin position (i.e. circular generalized Heegaard splittings). This is an idea that was introduced by Fabiola Manjarrez-Gutierrez for studying knot complements , though as she notes, it can be applied to any 3-manifold with infinite first homology. Alexander Coward gave a talk about using these ideas to study knots with unknotting number one (i.e. knots that become the unknot after a single crossing change) and he pointed out a difference between circular thin position and standard thin position that really blew me away: There are infinitely many circular generalized Heegaard splittings for the unknot that come from stabilizing the minimal thin position exactly once. Below the fold, I’ll give a brief description of circular thin position, then explain this surprising phenomenon.