Computation-free presentation of the fundamental group of generic $(p,q)$-torus curves

TL;DR

This paper introduces a novel, computation-free approach to determine the fundamental groups of curve complements, specifically applied to generic (p,q)-torus curves, simplifying previous methods.

Contribution

It presents a new variation of the Zariski-Van Kampen method on ruled surfaces, providing an alternative proof for the fundamental group of generic (p,q)-torus curves.

Findings

Provides a computation-free proof for the fundamental group of generic (p,q)-torus curves.

Introduces a new method using ruled surfaces for fundamental group calculations.

Simplifies the understanding of the topology of curve complements.

Abstract

In this note, we present a new method for computing fundamental groups of curve complements using a variation of the Zariski-Van Kampen method on general ruled surfaces. As an application we give an alternative (computation-free) proof for the fundamental group of generic $(p,q)$-torus curves.