Abstract

Heegaard Floer homology, HF, is a 3-manifold invariant defined combinatorially in terms of the Heegaard diagram for the manifold. There is also a version for knots in 3-manifolds called HFK . Embedded contact homology, ECH , is a dynamic invariant defined in terms of periodic orbits of the contact Reeb vector field, and by controlling orbits near the knot a knot version, ECK , can be obtained in this setting as well. Colin, Ghiggini and Honda recently proved that the 3-manifold invariants HF and ECH are isomorphic, and it is conjectured that HFK and ECK are isomorphic. With the help of pictures and examples, I will talk about these two theories, discuss their isomorphism and, if time permits, talk about my research concerning a surgery formula in ECK analogous to a well-known formula on HFK .