Geometry and Topology Seminar
Ozsvath and Szabo have shown that knot Floer homology detects knot genus - the largest Alexander grading of a non-trivial homology class is equal to the genus.
We give a new contact geometric interpretation of this fact by realizing such a class via the transverse knot invariant introduced by Lisca, Ozsvath, Stipsicz and Szabo. Our approach relies on the "convex decomposition theory" of Honda, Kazez and Matic - a contact geometric interpretation of Gabai's sutured hierarchies.
We use this new interpretation to study the "next-to-top" summand of knot Floer homology, and to show that Heegaard Floer homology detects quasi-positive Seifert surfaces. Some of this talk represents joint work with Matthew Hedden.