Multiplicity and lattice cohomology of plane curve singularities

The lattice cohomology and the graded root of an isolated curve singularity were recently introduced in [3]. The lattice cohomology is a categorification of the δ -invariant. The hope is that for plane curve singularities it encodes subtle information about the analytic structure and concrete analytic invariants. The present paper is a positive result in this direction: we prove that the multiplicity of an irreducible plane curve singularity can be recovered from its lattice cohomology (or, from its graded root). In fact, we give four distinct proofs of this statement, each of them emphasizing a rather different aspect of the theory of plane curve germs. With these proofs, we also create new bridges between the abstract analytic type and the embedded topological type of the germ. In particular, we provide a new characterization of the Ap´ery set of the semigroup of the germ in terms of embedded data.