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Series: Geometry Topology Seminar

We recall the Schur Weyl duality from representation theory and show how this can be applied to express the colored Jones polynomial of torus knots in an elegant way. We'll then discuss some applications and further extensions of this method.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

We will introduce new constructions of infinite families of smooth structures on small 4-manifolds and infinite families of smooth knottings of surfaces.

Series: Geometry Topology Seminar

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.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

Legendrian knots are knots that can be described only by their projections(without having to separately keep track of the over-under crossinginformation): The third coordinate is given as the slope of theprojections. Every knot can be put in Legendrian position in many ways. Inthis talk we present an ongoing project (with Etnyre and Ng) of thecomplete classification of Legendrian representations of twist knots.

Series: Geometry Topology Seminar

The uniform thickness property (UTP) is a property of knots embeddedin the 3-sphere with the standard contact structure. The UTP was introduced byEtnyre and Honda, and has been useful in studying the Legendrian and transversalclassification of cabled knot types. We show that every iterated torus knotwhich contains at least one negative iteration in its cabling sequence satisfiesthe UTP. We also conjecture a complete UTP classification for iterated torusknots, and fibered knots in general.

Series: Geometry Topology Seminar

A closed hyperbolic 3-manifold $M$ determines a fundamental classin the algebraic K-group $K_3^{ind}(C)$. There is a regulator map$K_3^{ind}(C)\to C/4\Pi^2Z$, which evaluated on the fundamental classrecovers the volume and Chern-Simons invariant of $M$. The definition of theK-groups are very abstract, and one is interested in more concrete models.The extended Bloch is such a model. It is isomorphic to $K_3^{ind}(C)$ andhas several interesting properties: Elements are easy to produce; thefundamental class of a hyperbolic manifold can be constructed explicitly;the regulator is given explicitly in terms of a polylogarithm.

Series: Geometry Topology Seminar

After reviewing a few techniques from the theory of confoliation in dimension three we will discuss some generalizations and certain obstructions in extending these techniques to higher dimensions. We also will try to discuss a few questions regarding higher dimensional confoliations.

Series: Geometry Topology Seminar