Combinatorial duality for Poincar\'e series, polytopes and invariants of plumbed 3-manifolds

TL;DR

This paper establishes a duality-based framework connecting Poincaré series, polytopes, and invariants of plumbed 3-manifolds, simplifying the computation of Seiberg-Witten invariants via lattice point counting and reciprocity principles.

Contribution

It introduces a novel duality approach that simplifies the calculation of Seiberg-Witten invariants using Ehrhart reciprocity and polytopal lattice point enumeration.

Findings

Derived a simple expression for the periodic constant using duality and Ehrhart reciprocity.

Generalized Seiberg-Witten invariants as multivariable polynomials related to lattice points.

Established a topological analogue of Khovanskii and Morales formulae for singularities.

Abstract

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 Poincar\'e 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 constant. 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.