Fundamental group of complex hypocycloids

Abstract

The Zariski–van Kampen theorem allows us to provide a presentation of the fundamental group for the complement of algebraic plane curves. However, certain computations require arduous work, as exemplified in the case of hypocycloids. In this paper we present the following result: Theorem 1. The fundamental group of any complex hypocycloid with N cusps is the Artin group of the N-gon. The main idea of the proof is take advantage of the symmetries inherent in the hypocycloid, allowing us to partition the domain to determine the generators of the fundamental group.