Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Configuration spaces of circles in the plane

Justin Curry, Ryan C. Gelnett, Matthew C. B. Zaremsky

TL;DR

This work studies spaces of circle configurations in the plane, revealing that each connected component is aspherical and has a fundamental group given by a braided automorphism group $\mathrm{bAut}(T)$ determined by the nesting tree $T$. The authors construct $\mathrm{bAut}(T)$ recursively as an iterated semidirect product of braid-group subgroups and show that the labeled case yields the pure braided automorphism group $\mathrm{pbAut}(T)$. The key methodology combines a circle-to-tree encoding with fiber-bundle arguments and deformation to disk configurations to establish $K(G,1)$-type spaces and explicit group identifications. The results connect to statistical mechanics, topological data analysis, and geometric group theory, and open avenues for extending the framework to infinite trees and other surfaces, enriching the study of braided symmetry in configuration spaces.

Abstract

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as "braided" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.

Configuration spaces of circles in the plane

TL;DR

This work studies spaces of circle configurations in the plane, revealing that each connected component is aspherical and has a fundamental group given by a braided automorphism group determined by the nesting tree . The authors construct recursively as an iterated semidirect product of braid-group subgroups and show that the labeled case yields the pure braided automorphism group . The key methodology combines a circle-to-tree encoding with fiber-bundle arguments and deformation to disk configurations to establish -type spaces and explicit group identifications. The results connect to statistical mechanics, topological data analysis, and geometric group theory, and open avenues for extending the framework to infinite trees and other surfaces, enriching the study of braided symmetry in configuration spaces.

Abstract

We consider the space of all configurations of finitely many (potentially nested) circles in the plane. We prove that this space is aspherical, and compute the fundamental group of each of its connected components. It turns out these fundamental groups are obtained as iterated semidirect products of subgroups of braid groups, with the structure for each component dictated by a finite rooted tree. These groups can be viewed as "braided" versions of the automorphism groups of such trees. We also discuss connections to statistical mechanics, topological data analysis, and geometric group theory.

Paper Structure

This paper contains 14 sections, 12 theorems, 32 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Connections: statistical mechanics and representation stability
  3. Connections: topological data analysis, deep learning and sensor networks
  4. Connections: geometric group theory
  5. Configuration spaces of points, circles, and disks
  6. Configurations of disks in the plane
  7. Between circle configurations and trees
  8. The labeled case
  9. The unlabeled case
  10. Connected components
  11. Braiding automorphism groups of finite trees
  12. Fundamental groups of configuration spaces of circles
  13. The unlabeled case
  14. The labeled case

Key Result

Theorem A

Let $T$ be a tree with $n$ non-root vertices. Then $\mathop{\mathrm{UConf}}\nolimits_T(S^1,\mathbb{R}^2)$ is aspherical, and its fundamental group is isomorphic to $\mathop{\mathrm{bAut}}\nolimits(T)$, hence it is a $K(\mathop{\mathrm{bAut}}\nolimits(T),1)$.

Figures (13)

  • Figure 1: Two configurations in different components of $\mathop{\mathrm{UConf}}\nolimits_2(S^1,\mathbb{R}^2)$.
  • Figure 2: The directory tree of React, a Javascript library, viewed as a configuration of nested circles; from CurranViz.
  • Figure 3: Two Embedded Spheres with the same Reeb Graph
  • Figure 4: Three embedded spheres. Projecting onto the $z$-axis yields a height function with 1 maximum, 3 saddles, and 4 minima in each example. Even with aligned critical values, they cannot be deformed to each other with a $z$-level set preserving isotopy.
  • Figure 5: An element of $\pi_1(\mathop{\mathrm{UConf}}\nolimits_4(*,\mathbb{R}^2))$, viewed as a braid.
  • ...and 8 more figures

Theorems & Definitions (36)

  • Theorem A
  • Definition A: Configuration space of points
  • Definition B: Configuration space of circles
  • Definition C: Configuration space of disks
  • Proposition D
  • proof
  • Remark E
  • Lemma F: Lemma 5.49 of idrissireal
  • proof
  • Corollary G
  • ...and 26 more