Arrangements of Pseudocircles: On Circularizability

TL;DR

This paper thoroughly investigates which arrangements of pseudocircles can be represented by actual circles, identifying all non-circularizable cases for small arrangements and using computational and geometric methods to prove non-circularizability.

Contribution

It provides the first comprehensive classification of circularizability for arrangements of pseudocircles with up to six elements, including new non-circularizable examples and proof techniques.

Findings

Exactly four non-circularizable arrangements of 5 pseudocircles.

Three non-circularizable arrangements among 2131 of 6 pseudocircles.

Eight additional arrangements of 6 pseudocircles proven non-circularizable.

Abstract

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set of 2131 digon-free intersecting arrangements of 6 pseudocircles we identify the three non-circularizable examples. We also show non-circularizability of 8 additional arrangements of 6 pseudocircles which have a group of symmetries of size at least 4. Most of our non-circularizability proofs depend on incidence theorems like Miquel's. In other cases we contradict circularizability by considering a continuous deformation where the circles of an assumed circle representation grow or shrink in a controlled way. The claims that we have all non-circularizable arrangements with the given properties are based on a program that generated all arrangements up to a certain size. Given the complete lists of arrangements, we used heuristics to find circle representations. Examples where the heuristics failed were examined by hand.