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We consider a connected negative definite plumbing graph, and we assume thatthe associated plumbed 3-manifold is a rational homology sphere. We provide twonew combinatorial formulae for the Seiberg-Witten invariant of this manifold.The first one is the constant term of a `multivariable Hilbert polynomial', itreflects in a conceptual way the structure of the graph, and emphasizes thesubtle parallelism between these topological invariants and the analyticinvariants of normal surface singularities. The second formula realizes theSeiberg-Witten invariant as the normalized Euler characteristic of the latticecohomology associated with the graph, supporting the conjectural connectionsbetween the Seiberg-Witten Floer homology, or the Heegaard-Floer homology, andthe lattice cohomology.

The Seiberg-Witten invariants of negative definite plumbed 3-manifolds