[Paper Review] On the topology of hypocycloids

Enrique Artal Bartolo, José Agustín|arXiv (Cornell University)|Mar 24, 2017

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper investigates the fundamental groups of complements of complex hypocycloids using symmetry, braid monodromy, and the Zariski-van Kampen method. It shows that for specific hypocycloids like the deltoid ($\rho = \frac{1}{3}$) and $\rho = \frac{2}{5}$, the fundamental groups are isomorphic to Artin groups of the pentagon and octagon, respectively, establishing a topological link between algebraic curves and braid-like group structures.

ABSTRACT

Algebraic geometry has many connections with physics: string theory, enumerative geometry, and mirror symmetry, among others. In particular, within the topological study of algebraic varieties physicists focus on aspects involving symmetry and non-commutativity. In this paper, we study a family of classical algebraic curves, the hypocycloids, which have links to physics via the bifurcation theory. The topology of some of these curves plays an important role in string theory and also appears in Zariski's foundational. We compute the fundamental groups of some of these curves and show that they are in fact Artin groups.

Motivation & Objective

• To determine the fundamental groups of the complements of complex hypocycloids in $\mathbb{C}^2$ and $\mathbb{P}^2$.
• To explore the topological structure of hypocycloids with rational curvature ratios, particularly focusing on their singularities and symmetries.
• To establish a connection between the topology of these curves and Artin groups through algebraic and geometric methods.
• To generalize the computation of fundamental groups for hypocycloids with $\rho = \frac{n}{2n+1}$, especially when all nodes are real.
• To develop a systematic method using quotient curves and symmetry reduction to simplify the computation of fundamental groups in complex algebraic geometry.

Proposed method

• Utilizes rational parametrization of hypocycloids via Chebyshev polynomials $T_n$, $U_n$, and $W_n$ to define complex and projective curves.
• Applies the Zariski-van Kampen method to compute the fundamental group of the complement of the projective hypocycloid $\bar{C}_{k,\ell}$.
• Employs braid monodromy techniques and Reidemeister-Schreier method to derive group presentations from geometric symmetries.
• Reduces the problem by quotienting the curve by the dihedral group $\mathbb{D}_{2N}$ action to obtain simpler, totally real quotient curves.
• Uses symmetry along coordinate axes (e.g., $x=0$, $y=0$) to further simplify the topology and make singularities visible in real projections.
• Validates results using computational algebra systems like GAP to verify group presentations and relations.

Experimental results

Research questions

• RQ1 What is the fundamental group of the complement of a complex hypocycloid $C_{k,\ell}$ in $\mathbb{C}^2$?
• RQ2 How do the symmetries of hypocycloids (especially dihedral symmetry $\mathbb{D}_{2N}$) influence the structure of their fundamental groups?
• RQ3 Can the fundamental group of a hypocycloid complement be identified as an Artin group, and if so, which one?
• RQ4 Under what conditions (e.g., rational $\rho = \ell/N$) are all nodes and singularities real, enabling topological simplification via quotienting?
• RQ5 Is there a general method to compute the fundamental group of $C_{n+1,n}$ using symmetry and quotient curves?

Key findings

• The fundamental group $\pi_1(\mathbb{C}^2 \setminus C_{3,1})$ for the deltoid ($\rho = \frac{1}{3}$) is isomorphic to the braid group on four strings, which is also the Artin group of the triangle.
• For $\rho = \frac{2}{5}$, the fundamental group $\pi_1(\mathbb{C}^2 \setminus C_{3,2})$ is isomorphic to the Artin group of the pentagon.
• For $\rho = \frac{3}{8}$, the fundamental group $\pi_1(\mathbb{C}^2 \setminus C_{5,3})$ is isomorphic to the Artin group of the octagon.
• The quotient curve $D_{k,\ell} = C_{k,\ell}/\mathbb{D}_{2N}$ is totally real when $\ell = k-1$, enabling topological simplification via real projections.
• The group $\pi_1(\mathbb{C}^2 \setminus C_{3,1})$ is not isomorphic to the braid group $\mathbb{B}_5$, correcting a prior claim in the literature.
• The method successfully recovers not only the correct group but also its correct presentation, suggesting a deep geometric link between symmetry and Artin group structure.