The theory of complex projective plane curves has a long history. However, curves of higher genus are rarely studied. It turns out that Heegaard-Floer theory can be effectively used to obtain constraints on possible cusp types of such curves. In fact, restricting ourselves to the case of curves with one cusp having a torus knot link, one can obtain an almost complete classification of possible torus knot types for infinitely many curve genera. The proof is a nice interplay of the theory of numerical semigroups, generalized Pell equations and birational transformations.
These results were obtained in a joint work with Daniele Celoria and Marco Golla. Independently, similar work was done by Maciej Borodzik, Matthew Hedden and Charles Livingston.