Abstract. We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.

We show a counterexample to a conjecture of de Bobadilla, Luengo, Melle-Hernández and Némethi on rational cuspidal projective plane curves, formulated in [3]. The counterexample is a tricuspidal curve of degree 8. On the other hand, we show that if the number of cusps is at most 2, then the original conjecture can be deduced from the recent results of Borodzik and Livingston ([5]) and the computations of [19].Then we formulate a 'simplified' (slightly weaker) version, more in the spirit of the motivation of the original conjecture (comparing index type numerical invariants), and we prove it for all currently known rational cuspidal curves. We make all these identities and inequalities more transparent in the language of lattice cohomology H * (S 3 −d (K)) of surgery 3-manifolds S 3 −d (K), where K = K1# · · · #Kν is a connected sum of algebraic knots. Finally, we prove that the zeroth lattice cohomology of this surgery manifold, H 0 (S 3 −d (K)) depends only on the multiset of multiplicities occurring in the multiplicity sequences describing the algebraic knots Ki. This result is closely related to the lattice-cohomological reformulation of the above mentioned theorems and conjectures, and provides new computational and comparison procedures.

In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant ν + : we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.

Abstract. Based on Tiankai Liu's PhD thesis [10], we give a complete classification of local topological types of singularities with two Newton pairs on rational unicuspidal complex projective plane curves.