On Poincaré series associated with links of normal surface singularities

Abstract:

Assume that $M$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal {T}$. We consider the topological Poincaré series associated with $\mathcal {T}$ and its counting functions, which encodes rich topological information, e.g., the Seiberg–Witten invariants of $M$.

In this article we study the counting functions via coefficient functions following the work of Szenes and Vergne. These are quasipolynomials on a special affine cone $\mathcal {S}’$ associated with the topology of $M$, in accordance with the previous results of Némethi and the first author. We prove that $\mathcal {S}’$ consists of a unique quasipolynomiality chamber, and we establish further structure theorems. We provide a formula for the counting function in terms of only one- and two-variable counting functions indexed by the edges and the vertices of the graph. This is the core of the proof for a “polynomial-negative degree part” decomposition theorem of the Poincaré series, which leads to a polynomial generalization of the Seiberg–Witten invariants of $M$. Finally, we reprove and discuss surgery formulas for the counting functions, in particular for the Seiberg–Witten invariants, using our methods.