Wirtinger curves, Artin groups, and hypocycloids

TL;DR

This paper introduces a Wirtinger presentation method for computing the fundamental group of algebraic plane curve complements, simplifying calculations and relating hypocycloids to Artin groups.

Contribution

It develops a new combinatorial Wirtinger presentation approach based on the real picture of curves, enabling explicit fundamental group computations for certain hypocycloids.

Findings

Computed fundamental groups for an infinite family of hypocycloids.

Established a connection between hypocycloids and Artin groups.

Provided a practical method to simplify algebraic curve group calculations.

Abstract

The computation of the fundamental group of the complement of an algebraic plane curve has been theoretically solved since Zariski-van Kampen, but actual computations are usually cumbersome. In this work, we describe the notion of Wirtinger presentation of such a group relying on the real picture of the curve and with the same combinatorial flavor as the classical Wirtinger presentation; we determine a significant family of curves for which Wirtinger presentation provides the required fundamental group. The above methods allow us to compute that fundamental group for an infinite subfamily of hypocycloids, relating them with Artin groups.