Schedule of Topology Seminars: 2009-10

Width is an invariant of knots that depends on the number of minima and maxima of the knot as well as their relative positions. The behavior of width with respect to connected sum has been studied for three decades. We want to know if width is additive under connected sum. I will outline classic results pertaining to the question and discuss ongoing joint work with Maggy Tomova that suggests width is not additive.

Let M be a closed 3-manifold with a given Heegaard splitting. We show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in the 3-sphere. Additionally we show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heeegaard splitting, and show that for Heegaard genus greater than or equal to 2 the coarse mapping class group of a Heegaard splitting is isomorphic to the genuine mapping class group of the Heegaard splitting. This is joint work with Matt Rathbun.

When a group acts on a simply connected CW complex, there is a way to recover a presentation (generators and relations) for the group in terms of stabilizers of cells. Given any commutative ring with one that is generated by shears, I will describe the construction of the 2nd Torus Complex over that ring, which is a connected simplicial complex. For certain rings (such as the Gaussian Integers and Eisensteinian Integers) I can prove that the complex is also simply connected.

An n-component link L in the three sphere has Generalized Property R provided that if L admits a 0-framed Dehn surgery yielding the n-fold connected sum of S^1 x S^2, then L is handleslide-equivalent to the 0-framed unlink (of n-components). We show that if the exterior of a 2-component link L in the three sphere admits a genus 2 Heegaard splitting, then L has Generalized Property R.

Given a 3-braid, we will discuss a method to find a representative word given its unreduced Burau matrix. The method used shows that every 3-braid is represented by a word of a nice form. We also discuss how this word informs the braid's positivity under the Dehornoy order.

I will give an overview of the main results in the 2009 PhD thesis of Aldo-Hilario Cruz-Cota. He determines the moduli space of singular Euclidean structures on the 2-sphere with 4 conepoints, two of angle 2pi/3 and two of angle 4pi/3. He gives an explicit parameterization of this moduli space.

About 10 years ago, Rubinstein and I gave a lengthy list of possible ways to construct Heegaard genus two manifolds with more than one isotopy class of Heegaard splittings. We also provided an argument why the list was complete (though possibly redundant). Recently John Berge noticed a missing case, one that is mathematically quite interesting. He was also able to prove cases were missing by exhibiting some of Heegaard distance 3, whereas all of ours were Heegaard distance 2. It appears that a rich mix of ideas was involved in his work, and I will talk about some that I understand.

We construct some embeddings of the torus into 4-space and study their properties. This talk is produced from discussions among Yi Liu, Yi Ni, Hongbin Sun and the speaker.