Knot invariants from Reeb orbits

Vincent Colin, Ko Honda, Michael Hutchings and I defined embedded contact homology groups for knots in a three-manifold as a slight modification of Hutching's embedded contact homology for closed three manifolds. I will sketch a strategy to prove that those groups are isomorphic to Ozsváth, Szabó and Rasmussen's knot Floer homology, and therefore are topological invariants. The strategy is to extend the knot complement to a larger closed manifold, and then apply the isomorphism between Heegaard Floer homology and embedded contact homology to the closed manifold. In the talk I will focus on the effect of that extension on  embedded contact homology, and therefore no knowledge of knot Floer homology will be necessary beyond the fact that it exists and is interesting. On the other hand the definition of embedded contact homology, both for knots and closed three-manifolds, will be sketched. This is a joint work in progress with Vincent Colin and Ko Honda.