On Poincar\'e series associated with links of normal surface singularities

TL;DR

This paper investigates topological Poincaré series linked to rational homology sphere plumbed 3-manifolds, revealing their structure, polynomial generalizations, and connections to Seiberg--Witten invariants through residue techniques.

Contribution

It introduces a new interpretation of Poincaré series as coefficient sums, proves quasipolynomiality in a specific cone, and constructs polynomial generalizations of Seiberg--Witten invariants.

Findings

Proves the uniqueness of quasipolynomiality inside a specific cone

Provides a graph-based structure for the counting function

Constructs polynomial generalizations of Seiberg--Witten invariants

Abstract

We study the counting function of topological Poincar\'e series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seiberg--Witten invariant given by the Poincar\'e series. We also reprove and discuss surgery formulas of N\'emethi for the counting function, and of Braun and N\'emethi for the Seiberg--Witten invariant.