Georgia Institute of Technology College of Sciences

I will discuss a conjecture that relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. I will present examples, as well as computational challenges.

In 3-dimensional contact topology one of the main problem is classifying Legendrian (transverse) knots in certain knot type up to Legendrian ( transverse) isotopy. In particular we want to decide if two (one in case of transverse knots) classical invariants of this knots are complete set of invariants. If it is, then we call this knot type Legendrian (transversely) simple knot type otherwise it is called Legendrian (transversely) non-simple. In this talk, by tracing the techniques developed by Etnyre and Honda, we will present some results concerning the complete Legendrian and transverse classification of certain cabled knots in the standard tight contact 3-sphere. Moreover we will provide an infinite family of Legendrian and transversely non-simple prime knots.

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

The deformation variety is similar to the representation variety inthat it describes (generally incomplete) hyperbolic structures on3-manifolds with torus boundary components. However, the deformationvariety depends crucially on a triangulation of the manifold: theremay be entire components of the representation variety which can beobtained from the deformation variety with one triangulation but notanother, and it is unclear how to choose a "good" triangulation thatavoids these problems. I will describe the "extended deformationvariety", which deals with many situations that the deformationvariety cannot. In particular, given a manifold which admits someideal triangulation we can construct a triangulation such that we canrecover any irreducible representation (with some trivial exceptions)from the associated extended deformation variety.