# The intrinsic topological nature of the Poincar\'e series of a plane curve singularity

## TL;DR

This paper explores the algebraic and topological properties of the Poincaré series of plane curve singularities, revealing its intrinsic connection to the link's Alexander polynomial through factorization and resolution graph analysis.

## Contribution

It introduces new factorization theorems for the Poincaré series, providing an algebraic method to compute it and offering a novel proof of its equality with the Alexander polynomial.

## Key findings

- Factorization theorems for Poincaré series based on semigroup values
- Algebraic computation of Poincaré series from dual resolution graph
- Proof of the coincidence between Poincaré series and Alexander polynomial

## Abstract

In this paper we provide some factorization theorems of the Poincar\'e series $P_C$ of a plane curve singularity $C$ depending on some key values of the semigroup of values of \(C\). These results yield an iterative computation of $P_C$ in purely algebraic terms from the dual resolution graph of $C$. On the other hand, Campillo, Delgado and Gusein-Zade showed in 2003 the equality between $P_C$ and the Alexander polynomial $\Delta_L$ of the corresponding link $L$. Our procedure supplies a new proof of this coincidence. More concretely, we show that our algebraic construction can be translated to the iterated toric structure of the link $L$. Additionally we show that the semigroup algebra can be defined from the fundamental group of the link exterior in the irreducible case. This gives in particular a conceptual reason for the coincidence of $P_C$ and $\Delta_L$.