Georgia Institute of Technology College of Sciences

The first hour of this talk gives a gentle introduction to yet another version of Heegaard Floer homology; Sutured Floer homology. This is the generalization of Heegaard Floer homology, for 3-manifolds with decorations (sutures) on their boundary. Sutures come naturally for contact 3-manifolds. Later we will concentrate on invariants for contact 3--manifolds in Heegaard Floer homology. This can be defined both for closed 3--manifolds, in this case they live in Heegaard Floer homology and for 3--manifolds with boundary, when the invariant is in sutured Floer homology. There are two natural generalizations of these invariants for Legendrain knots. One can directly generalize the definition of the contact invariant $\widehat{\mathcal{L}}$, or one can take the complement of the knot, and compute the invariant for that:$\textrm{EH}$. At the end of this talk I would like to describe a map that sends $\textrm{EH}$ to$\widehat{\mathcal{L}}$. This is a joint work with Andr\'as Stipsicz.

Pseudo-Anosov mapping classes on surfaces have a rich structure, uncovered by William Thurston in the 1980's. We will discuss the 1995 Bestvina-Handel algorithmic proof of Thurston's theorem, and in particular the "transition matrix" T that their algorithm computes. We study the Bestvina-Handel proof carefully, and show that the dilatation is the largest real root of a particular polynomial divisor P(x) of the characteristic polynomial C(x) = | xI-T |. While C(x) is in general not an invariant of the mapping class, we prove that P(x) is. The polynomial P(x) contains the minimum polynomial M(x) of the dilatation as a divisor, however it does not in general coincide with M(x).In this talk we will review the background and describe the mathematics that underlies the new invariant. This represents joint work with Peter Brinkmann and Keiko Kawamuro.

Given a knot, a simple Lie algebra L and an irreducible representation V of L one can construct a one-variable polynomial with integer coefficients. When L is the simplest simple Lie algebra (sl_2) this gives a sequence of polynomials, whose sequence of degrees is a quadratic quasi-polynomial. We will discuss a conjecture for the degree of the colored Jones polynomial for an arbitrary simple Lie algebra, and we will give evidence for sl_3. This is joint work with Thao Vuong.

There are many conjectured connections between Heegaard Floer homology and the various homologies appearing in low dimensional topology and symplectic geometry. One of these conjectures states, roughly, that if \phi is a diffeomorphism of a closed Riemann surface, a certain portion of the Heegaard Floer homology of the mapping torus of \phi should be equal to the Symplectic Floer homology of \phi. I will discuss how this can be confirmed when \phi is periodic (i.e., when some iterate of \phi is the identity map). I will recall how a mapping torus can be realized via Dehn surgery; then, I will sketch how the surgery long exact triangles of Heegaard Floer homology can be distilled into more direct surgery formulas involving knot Floer homology. Finally, I'll say a few words about what actually happens when you use these formulas for the aforementioned Dehn surgeries: a "really big game of tic-tac-toe".