Abstract: Ozsvath and Szabo, as part of the development of Heegaard Floer Homology, created a contact invariant that exhibits some extremely nice properties related to foliations and symplectic 4-manifolds. One way to see the invariant is to use a Heegaard Diagram for a 3-manifold coming from a compatible open book decomposition. This perspective, originally due to Honda, Kazez and Matic, gives us a strong geometric connection between the contact invariant and the monodromy of the open book decomposition. We will discuss the HKM construction, present many examples, and discuss recent progress on an effort to make the contact invariant more sensitive to whether the contact structure is tight or overtwisted. This is joint work with Cagatay Kutluhan, Gordana Matic and Andy Wand.
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