N\'emethi's division algorithm for zeta-functions of plumbed 3-manifolds

TL;DR

This paper introduces an algorithm based on multivariable Euclidean division to explicitly compute a polynomial invariant related to Seiberg-Witten invariants for negative definite plumbed 3-manifolds, advancing computational methods in topology.

Contribution

It presents a novel algorithm for calculating the polynomial invariant associated with plumbed 3-manifolds using Euclidean division of the zeta-function.

Findings

Algorithm enables explicit computation of the polynomial invariant.

Facilitates calculation of Seiberg-Witten invariants for specific 3-manifolds.

Provides a practical tool for topological invariants analysis.

Abstract

A polynomial counterpart of the Seiberg-Witten invariant associated with a negative definite plumbed 3-manifold has been proposed by earlier work of the authors. It is provided by a special decomposition of the zeta-function defined by the combinatorics of the manifold. In this article we give an algorithm, based on multivariable Euclidean division of the zeta-function, for the explicit calculation of the polynomial, in particular for the Seiberg--Witten invariant.