possible configurations of ovals (bounded connected components); see, e.g., [@Viro]. However, the curves $R$ and $I$ are *harmonic*, and so the maximum principle [@Ahlfors Theorem 21, p. 166] implies that they have no ovals at all. The topological information that can be extracted from the pair $R,I$ is of a different sort entirely.

Suppose that $R$ and $I$ are both nonsingular. (In particular, it is necessary that $f(z)$ have no repeated roots; as we will see, this condition is in a certain sense sufficient.) Labeling the $2n$ points of $R\cap S$ cyclically and pairing off those points that lie in the same component of $R$ yields a *noncrossing matching* of order $n$. (Noncrossing partitions are an important subject in modern combinatorics; for an overview, see the excellent survey by Simion [@Simion].) Repeating the construction for $I$ instead of $R$ yields a second noncrossing matching that is interlaced with the first. The matching obtained from $R$ must cross the matching obtained from $I$ exactly $n$ times, corresponding to the $n$ roots of the polynomial $f$. Two arbitrary interlaced noncrossing matchings of order $n$ must have at least $n$ crossings between them (Lemma \[lemma:odd-cross\]), and may have more; if the number of 