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

By A. N'emethi. 2010

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Assume that0 is a connected negative definite plumbing graph, and that the associated plumbed 3-manifoldM is a rational homology sphere. We provide two new combinatorial formu- lae for the Seiberg-Witten invariant of M. The first one is the constant term of a 'multivariable Hilbert polynomial', it reflects in a conceptual way the structure of the graph 0, and emphasizes the subtle parallelism between these topological invariants and the analytic invariants of normal surface singularities. The second formula realizes the Seiberg-Witten invariant as the normalized Euler characteristic of the lattice cohomology associated with 0, supporting the conjectural con- nections between the Seiberg-Witten Floer homology, or the Heegaard-Floer homology, and the lattice cohomology.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

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