Combinatorial duality for Poincaré series, polytopes and invariants of plumbed 3-manifolds
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Assume that the link of a complex normal surface singularity is a rational homology sphere. Then its Seiberg-Witten invariant can be computed as the ‘periodic constant’ of the topological multivariable Poincaré series (zeta function). This involves a complicated regularization procedure (via quasipolynomials measuring the asymptotic behaviour of the coefficients). We show that the (a Gorenstein type) symmetry of the zeta function combined with Ehrhart–Macdonald–Stanley reciprocity (of Ehrhart theory of polytopes) provide a simple expression for the periodic cosntant. Using these dualities we also find a multivariable polynomial generalization of the Seiberg–Witten invariant, and we compute it in terms of lattice points of certain polytopes. All these invariants are also determined via lattice point counting, in this way we establish a completely general topological analogue of formulae of Khovanskii and Morales valid for singularities with non-degenerate Newton principal part.