Seminars and Colloquia by Series

The Jones slopes of a knot

Series
Geometry Topology Seminar
Time
Monday, November 30, 2009 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros GaroufalidisGeorgia Tech
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.

Geometry, computational complexity and algebraic number fields

Series
Geometry Topology Seminar
Time
Monday, November 23, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Hong-Van LeMathematical Institute of Academy of Sciences of the Czech Republic
In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.

On the Legendrian and transverse classification of cabled knot types

Series
Geometry Topology Seminar
Time
Monday, November 9, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Bulent TosunGa Tech
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.

Schur Weyl duality and the colored Jones polynomial

Series
Geometry Topology Seminar
Time
Wednesday, October 28, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 255
Speaker
Roland van der VeenUniversity of Amsterdam
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.

Triple linking numbers, Hopf invariants and integral formulas for 3-component links

Series
Geometry Topology Seminar
Time
Monday, October 26, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Shea Vela-VickColumbia University
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.

A generalisation of the deformation variety

Series
Geometry Topology Seminar
Time
Monday, October 12, 2009 - 14:05 for 2 hours
Location
Skiles 269
Speaker
Henry SegermanUTexas Austin
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.

Classification of Legendrian twist knots

Series
Geometry Topology Seminar
Time
Monday, September 28, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Vera VertesiMSRI
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.

The uniform thickness property and iterated torus knots

Series
Geometry Topology Seminar
Time
Monday, September 21, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Doug LaFountainSUNY - Buffalo
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.

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