Arrangements of circles, the regions surrounded by them and labeled Poincar\'e-Reeb graphs

TL;DR

This paper explores the structure of regions formed by circle arrangements using Poincaré-Reeb graphs, introducing labels and analyzing local changes, with connections to real algebraic maps and singularity theory.

Contribution

It introduces labeled Poincaré-Reeb graphs for circle arrangements and studies their local modifications, extending previous work and linking to algebraic map singularities.

Findings

Labeled Poincaré-Reeb graphs encode circle arrangement regions.

New types of local changes in these graphs are identified.

Regions correspond to images of real algebraic maps, generalizing sphere projections.

Abstract

We are interested in arrangements of circles and the regions surrounded by them. {\it Poincar\'e-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 Poincar\'e-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.