Georgia Institute of Technology College of Sciences

Certain fibered hyperbolic 3-manifolds admit a layered veering triangulation, which can be constructed algorithmically given the stable lamination of the monodromy. These triangulations were introduced by Agol in 2011, and have been further studied by several others in the years since. We present experimental results which shed light on the combinatorial structure of veering triangulations, and its relation to certain topological invariants of the underlying manifold.

We give a simple geometric criterion for an element to normally generate the mapping class group of a surface. As an application of this criterion, we show that when a surface has genus at least 3, every periodic mapping class except for the hyperelliptic involution normally generates. We also give examples of pseudo-Anosov elements that normally generate when genus is at least 2, answering a question of D. Long.

This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.

Defined in the early 2000's by Ozsvath and Szabo, Heegaard Floer homology is a package of invariants for three-manifolds, as well as knots inside of them. In this talk, we will describe how work from Poul Heegaard's 1898 PhD thesis, namely the idea of a Heegaard splitting, relates to the definition of this invariant. We will also provide examples of the kinds of questions that Heegaard Floer homology can answer. These ideas will be the subject of the topics course that I am teaching in Fall 2017.

Continuing from last time, we will discuss Hilden and Montesinos' result that every smooth closed oriented three manifold is a three fold branched cover over the three sphere, and also there is a representation by bands.

Let S be a Riemann surface of type (p,1), p > 1. Let f be a point-pushing pseudo-Anosov map of S. Let t(f) denote the translation length of f on the curve complex for S. According to Masur-Minsky, t(f) has a uniform positive lower bound c_p that only depends on the genus p.Let F be the subgroup of the mapping class group of S consisting of point-pushing mapping classes. Denote by L(F) the infimum of t(f) for f in F pseudo-Anosov. We know that L(F) is it least c_p.

In this series of talks we will show that every closed oriented three manifold is a branched cover over the three sphere, with some additional properties. In the first talk we will discuss some examples of branched coverings of surfaces and three manifolds, and a classical result of Alexander, which states that any closed oriented combinatorial manifold is always a branched cover over the sphere.

In this talk we associate a combinatorial dg-algebra to a cubic planar graph. This algebra is defined by counting binary sequences, which we introduce, and we shall provide explicit computations. From there, we study the Legendrian surfaces behind these combinatorial constructions, including Legendrian surgeries and the count of Morse flow trees, and discuss the proof of the correspondence between augmentations and constructible sheaves for this class of Legendrians.

I will describe a diagrammatic classification of (1,1) knots in S^3 and lens spaces that admit non-trivial L-space surgeries. A corollary of the classification is that 1-bridge braids in these manifolds admit non-trivial L-space surgeries. This is joint work with Sam Lewallen and Faramarz Vafaee.