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

TL;DR

This paper introduces two new combinatorial formulas for calculating the Seiberg-Witten invariants of negative definite plumbed 3-manifolds, linking topological, analytic, and homological perspectives.

Contribution

It provides novel combinatorial methods to compute Seiberg-Witten invariants, connecting graph structures with lattice cohomology and Floer homology theories.

Findings

Two new formulas for Seiberg-Witten invariants

Connection between topological invariants and surface singularities

Support for conjectural links between Floer and lattice cohomology

Abstract

We consider a connected negative definite plumbing graph, and we assume that the associated plumbed 3-manifold is a rational homology sphere. We provide two new combinatorial formulae for the Seiberg-Witten invariant of this manifold. The first one is the constant term of a `multivariable Hilbert polynomial', it reflects in a conceptual way the structure of the graph, 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 the graph, supporting the conjectural connections between the Seiberg-Witten Floer homology, or the Heegaard-Floer homology, and the lattice cohomology.