Heegaard Floer homology, knotifications of links, and plane curves with noncuspidal singularities

We describe a formula for the H₁–action on the knot Floer homology of knotifications of links in S³. Using our results about knotifications, we are able to study complex curves with noncuspidal singularities, which were inaccessible using previous Heegaard Floer techniques. We focus on the case of a transverse double point, and give examples of complex curves of genus g which cannot be topologically deformed into a genus g 1 surface with a single double point.