We apply the methods of Heegaard Floer homology to identify topological properties of complex curves in CP 2 . As one application, we resolve an open conjecture that constrains the Alexander polynomial of the link of the singular point of the curve in the case that there is exactly one singular point, having connected link, and the curve is of genus zero. Generalizations apply in the case of multiple singular points.

We give a complete classification of algebraic curves in C 2 which are homeomorphic with C * and which satisfy a certain natural condition about codimensions of its singularities. In the proof we use the method developed in [BZI]. It relies on estimation of certain invariants of the curve, the so-called numbers of double points hidden at singularities and at infinity. The sum of these invariants is given by the Poincaré-Hopf formula applied to a suitable vector field.

Abstract. We compute the Heegaard Floer homology of S 3 1 (K) (the (+1) surgery on the torus knot Tp,q) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsváth-Szabó d-invariant. We relate the result to known knot invariants of Tp,q as the genus and the Levine-Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard Floer homologies of (+1) and (−1) surgeries on torus knots. This relation is best seen at the level of τ functions.