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 Gr\"unbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Gr\"unbaum 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 with $p_3(\mathcal{A})/n \to 16/11 = 1.\overline{45}$. We expect that the lower bound $p_3(\mathcal{A}) \geq 4n/3$ 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 $p_3 \geq 2n/3$, and conjecture that $p_3 \geq n-1$. Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that $p_3 \le 2n^2/3 +O(n)$. This is essentially best possible because there are families of pairwise intersecting arrangements of $n$ pseudocircles with $p_3/n^2 \to 2/3$. 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.