Arrangements of Pseudocircles: Triangles and Drawings

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 p3 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 with p3(A)/n→ 16/11 = 1.45. We expect that the lower bound p3(A) ≥ 4n/3 is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of n pairwise intersecting circles indeed have at least 2n− 4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of p3 ≥ 2n/3, and conjecture that p3 ≥ n− 1. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p3 ≤ 2n/3 + O(n). This is essentially best possible because families of pairwise intersecting arrangements of n pseudocircles with p3/n 2 → 2/3 as n→∞ are known. 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 the generation algorithm. In the final section we describe some aspects of the drawing algorithm.