| In the first part, we review the Floer homology theory constructed by Ozsvath and Szabo, in particular the Heegaard Floer homology of a knot K ⊂ S3. We compute these homology groups for the pretzel knots P(-2 a, 2b + 1, 2c + 1) and P(2a, -2b - 1, 2 c + 1), where a, b, c are positive integers. In particular, we re-prove that the slice genus and the unknotting number of P(-2a, 2 b + 1, 2c + 1) are b + c + 1. Then we introduce a Floer homology theory HFL&d14;K assigned to a knot K ⊂ S 3, parallel to HFK&d14;K , and derive some basic properties of this Floer theory. In particular we show that HFL&d14;K distinguishes the genus of K. We show that the Ozsvath-Szabo homology group HFK&d14;KL , of the Whitehead double KL of K, is isomorphic to HFL&d14;K .;In the second part, we review the Gromov-Witten invariants, as introduced by Ruan and Tian, together with the construction of Gromov invariants by Taubes. We combine these ideas to introduce the invariants Ig:P2X,Z &rarrr; Z, which assign integers to prime homology classes of the symplectic threefold (X, o), for any genus g > 1. We expect to extend this construction to get the invariants Ig:H2X,Z &rarrr; Z, defined for any homology class b∈H2X,Z . There is some relation with the Gopakumar-Vafa conjecture. |
