Vincent Colin, Ko Honda, Michael Hutchings and I defined embedded contact homo-
logy 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 the-
refore 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.
