such crossings is exactly $n$, we call this pair of noncrossing matchings a *basketball of order $n$*. For example, the right-hand side of Figure \[basketball-example-figure\] is the basketball $$\begin{aligned}
  \big( & \{\{0,10\},\{2,8\},\{4,6\},\{12,18\},\{14,16\}\},\\
  & \{\{1,19\},\{3,5\},\{7,9\},\{11,13\},\{15,17\}\}\big).
  \end{aligned}$$

It is natural to ask whether every basketball arises from a polynomial in this way. Our main result, the Inverse Basketball Theorem (Theorem \[IBT\]), answers this question in the affirmative. The proof is constructive, and draws on elementary tools from combinatorics, topology and complex analysis.

There is a natural bijection between basketballs of order $n$ and noncrossing partitions with $n$ blocks of size 4. It follows from a result of Edelman [@Edelman] that the basketballs are enumerated by the quasi-Catalan numbers $\frac{1}{3n+1}\binom{4n}{n}$. These numbers also count plane quaternary trees and dissections of a polygon into pentagons; there are natural bijections between basketballs and each of these combinatorial sets. The quasi-Catalan numbers also enumerate certain plane trees (see [@LW]), although there is no obvious bijection involved.

A more subtle combinatorial-topological invariant of the polynomial $f$ arises from the curves $${C_{\theta}}(f) = \{ z {\;:~}{\mathop{\rm Im}\nolimits}(e^{-i\theta} f(z)) = 0\},$$ regarded as a family parameterized by $\theta\in{{\mathbb R}}/\pi{{\mathbb Z}}$. It turns out that $C_\theta(f)$ is singular for only 