Seminars and Colloquia by Series

The degree of the colored Jones polynomial of a knot

Series
Geometry Topology Seminar
Time
Monday, October 11, 2010 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Stavros GaroufalidisGeorgia Tech
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.

Legendrian contact homology for Seifert fibered spaces

Series
Geometry Topology Seminar
Time
Monday, October 4, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Joan LicataStanford University
In this talk, I'll focus on Seifert fibered spaces whose fiber structure is realized by the Reeb orbits of an appropriate contact form. I'll describe a rigorous combinatorial formulation of Legendrian contact homology for Legendrian knots in these manifolds. This work is joint with J. Sabloff.

Surgery Formulas and Heegaard Floer Homology of Mapping Tori

Series
Geometry Topology Seminar
Time
Monday, September 27, 2010 - 17:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Evan FinkUniversity of Georgia

Please Note: This is the second talk in the Emory-Ga Tech-UGA joint seminar. The first talk will begin at 3:45.

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".

HOMFLY-PT polynomial and Legendrian links in the solid torus

Series
Geometry Topology Seminar
Time
Monday, September 27, 2010 - 15:45 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dan RutherfordDuke University

Please Note: This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will follow at 5.

A smooth knot in a contact 3-manifold is called Legendrian if it is always tangent to the contact planes. In this talk, I will discuss Legendrian knots in R^3 and the solid torus where knots can be conveniently viewed using their `front projections'. In particular, I will describe how certain decompositions of front projections known as `normal rulings' (introduced by Fuchs and Chekanov-Pushkar) can be used to give combinatorial descriptions for parts of the HOMFLY-PT and Kauffman polynomials. I will conclude by discussing recent generalizations to Legendrian solid torus links. It is usual to identify the `HOMFLY-PT skein module' of the solid torus with the ring of symmetric functions. In this context, normal rulings can be used to give a knot theory description of the standard scalar product determined by taking the Schur functions to form an orthonormal basis.

A monodromy invariant in the space of knots

Series
Geometry Topology Seminar
Time
Monday, September 13, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Jason McGibbonUniversity of Massachusetts
Knot contact homology (KCH) is a combinatorially defined topological invariant of smooth knots introduced by Ng. Work of Ekholm, Etnyre, Ng and Sullivan shows that KCH is the contact homology of the unit conormal lift of the knot. In this talk we describe a monodromy result for knot contact homology,namely that associated to a path of knots there is a connecting homomorphism which is invariant under homotopy. The proof of this result suggests a conjectural interpretation for KCH via open strings, which we will describe.

Spherical images of hypersurfaces

Series
Geometry Topology Seminar
Time
Monday, August 30, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Mohammad GhomiGa Tech
We discuss necessary and sufficient conditions of a subset X of the sphere S^n to be the image of the unit normal vector field (or Gauss map) of a closed orientable hypersurface immersed in Euclidean space R^{n+1}.

Knots in overtwisted contact structures

Series
Geometry Topology Seminar
Time
Monday, August 23, 2010 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
John EtnyreGa Tech
The study of Legendrian and transversal knots has been an essential part of contact topology for quite some time now, but until recently their study in overtwisted contact structures has been virtually ignored. In the past few years that has changed. I will review what is know about such knots and discuss recent work on the "geography" and "botany" problem.

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