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Arrangements of circles, the regions surrounded by them and labeled Poincaré-Reeb graphs

Naoki Kitazawa

TL;DR

The paper studies regions bounded by circles through the lens of Poincaré-Reeb graphs, enriching these graphs with labels that encode how the circles sit inside the regions. Starting from normally inductive NI arrangements of circles, it develops labeled Poincaré-Reeb V-digraphs relative to the projections ${\pi}_{2,1,i}$, detailing edge and vertex labeling rules that track tangent data and circle-arcs. It then analyzes local graph changes when new circles are added, proving results that classify how the labeled graphs mutate (including MBCC and SSC-NI contexts). The work links these combinatorial objects to explicit real algebraic maps and singularity theory, and proposes a categorical framework and explicit realizations of labeled PR graphs. Overall, it provides a concrete, label-aware framework to understand the topology of circle-based regions and their algebraic origins.

Abstract

We are interested in arrangements of circles and the regions surrounded by them. {\it Poincaré-Reeb graphs} have been fundamental and strong tools in studying shapes of regions surrounded by real algebraic curves, since around 2020. They are natural graphs the regions naturally collapse to and were first formulated by Sorea with several researchers. Studying shapes of such regions is one of fundamental studies in real algebraic geometry and combinatorics for example. This is surprisingly new and recently developing. Our study introduces labels on vertices and edges of such graphs encoding information of the circles where we concentrate on regions surrounded by circles. The author studied local changes of Poincaré-Reeb graphs by addition of circles under certain rules before and we discuss changes of new types. The author has started related studies motivated by singularity theory of real algebraic maps and found first that our regions are the images of natural real algebraic maps, generalizing natural projections of spheres.

Arrangements of circles, the regions surrounded by them and labeled Poincaré-Reeb graphs

TL;DR

The paper studies regions bounded by circles through the lens of Poincaré-Reeb graphs, enriching these graphs with labels that encode how the circles sit inside the regions. Starting from normally inductive NI arrangements of circles, it develops labeled Poincaré-Reeb V-digraphs relative to the projections , detailing edge and vertex labeling rules that track tangent data and circle-arcs. It then analyzes local graph changes when new circles are added, proving results that classify how the labeled graphs mutate (including MBCC and SSC-NI contexts). The work links these combinatorial objects to explicit real algebraic maps and singularity theory, and proposes a categorical framework and explicit realizations of labeled PR graphs. Overall, it provides a concrete, label-aware framework to understand the topology of circle-based regions and their algebraic origins.

Abstract

We are interested in arrangements of circles and the regions surrounded by them. {\it Poincaré-Reeb graphs} have been fundamental and strong tools in studying shapes of regions surrounded by real algebraic curves, since around 2020. They are natural graphs the regions naturally collapse to and were first formulated by Sorea with several researchers. Studying shapes of such regions is one of fundamental studies in real algebraic geometry and combinatorics for example. This is surprisingly new and recently developing. Our study introduces labels on vertices and edges of such graphs encoding information of the circles where we concentrate on regions surrounded by circles. The author studied local changes of Poincaré-Reeb graphs by addition of circles under certain rules before and we discuss changes of new types. The author has started related studies motivated by singularity theory of real algebraic maps and found first that our regions are the images of natural real algebraic maps, generalizing natural projections of spheres.

Paper Structure

This paper contains 14 sections, 10 theorems, 1 figure.

Table of Contents

  1. Introduction.
  2. Notation on topological spaces and manifolds.
  3. Our arrangements of circles.
  4. Organization of our paper and our main work.
  5. Topological spaces decomposed into cells, smooth maps and graphs.
  6. Poincaré-Reeb V-digraphs with labels.
  7. Reviewing the Poincaré-Reeb V-digraph of $(\mathcal{S},D_{\mathcal{S}})$ for ${\pi}_{2,1,i}$.
  8. The Poincaré-Reeb V-digraph of $(\mathcal{S},D_{\mathcal{S}})$ for ${\pi}_{2,1,i}$ with labels.
  9. Labels to edges.
  10. For labels to vertices of the graph.
  11. Our summary on the labels.
  12. Local changes of Poincaré-Reeb V-digraphs of $(\mathcal{S},D_{\mathcal{S}})$ of a certain new type according to additions of circles.
  13. Additional remarks.
  14. Conflict of interest and Data availability.

Key Result

Proposition 1

We concentrate on the case $S_{e,1}=S_{e,2}$. In the case $i=1$ ($i=2$ ), for elements in the first set here, it must hold that $4 \leq j^{\prime} \leq 7$ (resp. $j^{\prime}=2,3,4,5$ ) and for elements in the second set here, it must hold that $0 \leq j^{\prime} \leq 3$ (resp. $j^{\prime}=0,1,6,7$ )

Figures (1)

  • Figure 1: The unit disk $D^2$ and the boundary $S^1$. Black dots show the $(j,\frac{\pi}{4})$-poles of the circle $S^1$ with $j=0,2,4,6$. Blue dots show the $(j,\frac{\pi}{4})$-poles of the circle $S^1$ with $j=1,3,5,7$. The (value of the) slopes of the red straight lines and straight dotted lines are same and they are positive and smaller than $1$. Each colored dotted line corresponds to each case of Theorem \ref{['thm:2']} (\ref{['thm:2.1.1']}), (\ref{['thm:2.1.2']}), (\ref{['thm:2.1.3']}), (\ref{['thm:2.1.4']}), and (\ref{['thm:2.1.5']}), from the bottom to the top. The red straight dotted lines are tangent lines to the points of the circle $S^1$.

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Theorem 1
  • Theorem 2
  • proof
  • ...and 1 more