Crossings between Curves with Many Tangencies

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be two families of two-way infinite x-monotone curves, no three of which pass through the same point. Assume that every curve in $\mathcal{A}$ lies above every curve in $\mathcal{B}$ and that there are m pairs of curves, one from $\mathcal{A}$ and the other from $\mathcal{B}$, that are tangent to each other. Then the number of proper crossings among the members of $\mathcal A\cup\mathcal B$ is at least (1/2 − o(1))m ln m. This bound is almost tight.