Chord diagrams, topological quantum field theory, and the sutured
Floer homology of solid tori
Abstract: I will talk about recent investigations of contact elements in the sutured Floer homology of solid tori, as part of the (1+1)-dimensional TQFT defined by Honda-Kazez-Matic. The Z_2 sutured Floer homology vector spaces in this case form a "categorification of Pascal's triangle", a triangle of vector spaces, with contact elements corresponding to chord diagrams and forming distinguished subsets of order given by the Narayana numbers. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams --- so that this talk is actually very elementary.
I will show that there are natural "creation and annihilation
operators" which allow us to define a QFT-type basis consisting of
contact elements; and contact elements are in bijective correspondence
with comparable pairs of basis elements with respect to a certain
partial order, and in a natural and explicit way. I will explain how
we can use this to extend Honda's notion of contact category to a
2-category, and possibly a 3-category.