Arrangements of Pseudocircles: Triangles and Drawings

Abstract

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells p_3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least 2n-4. We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with p_3({mathscr {A}})/n rightarrow 16/11 = 1.overline{45}. We expect that the lower bound p_3({mathscr {A}}) ge 4n/3 is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of n pairwise intersecting circles have at least 2n-4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p_3 ge 2n/3, and conjecture that p_3 ge n-1. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p_3 le frac{4}{3}left( {begin{array}{c}n 2end{array}}right) +O(n). This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with p_3 = frac{4}{3}left( {begin{array}{c}n 2end{array}}right) . The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.