# Surgery formulae for the Seiberg-Witten invariant of plumbed 3-manifolds

## TL;DR

This paper derives surgery formulae for the Seiberg-Witten invariants of plumbed 3-manifolds using combinatorial series and their periodic constants, linking topological changes to algebraic invariants.

## Contribution

It introduces a method to compute Seiberg-Witten invariants via the periodic constant of a multivariable Poincaré series, providing explicit surgery formulae.

## Key findings

- Surgery formulae relate Seiberg-Witten invariants before and after vertex removal.
- Periodic constant encodes the difference in invariants for modified manifolds.
- Method applies to rational homology sphere plumbed 3-manifolds.

## Abstract

Assume that $M(\mathcal{T})$ is a rational homology sphere plumbed 3-manifold associated with a connected negative definite graph $\mathcal{T}$. We consider the combinatorial multivariable Poincar\'e series associated with $\mathcal{T}$ and its counting functions, which encode rich topological information. Using the `periodic constant' of the series (with reduced variables) we prove surgery formulae for the normalized Seiberg-Witten invariants: the periodic constant appears as the difference of the Seiberg-Witten invariants associated with $M(\mathcal{T})$ and $M(\mathcal{T}\setminus\mathcal{I})$, where $\mathcal{I}$ is an arbitrary subset of the set of vertices of $\mathcal{T}$.