HOMFLY-PT polynomial and Legendrian links in the solid torus

Series: 
Geometry Topology Seminar
Monday, September 27, 2010 - 15:45
1 hour (actually 50 minutes)
Location: 
Skiles 269
,  
Duke University
Organizer: 

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.