to half-rotation and *reflection*, replacing $\{i,j\},\{k,\ell\}$ with $\{-i,-j\},\{-k,-\ell\}$.

3.  It would be interesting to enumerate the bimatchings of order $n$ by total number of crossings.

The Inverse Basketball Theorem {#section:IBT}
==============================

In light of Gauss’s proof of the FTA, it is natural to ask whether every basketball of order $n$ arises as $B(f,0,\Pi/2)$ for some suitably chosen polynomial $f(z)$ of degree $n$. The main result of this section is that something more general is true.

\[IBT\] Fix any $\alpha,\beta \in [0,\pi)$ with $\alpha < \beta$. Then every basketball has the form ${B}(f,\alpha,\beta)$ for some monic polynomial $f$.

Before plunging into the details, let us describe the argument informally. Proposition \[thm:ear-thm\] suggests an inductive approach. Given an $n$-basketball $B$ for which we want to construct a realizing polynomial, we would like to remove an ear from $B$, inductively construct a realizing polynomial $f(z)$ for the resulting $(n-1)$-basketball $B'$, and then insert the missing ear by replacing $f(z)$ with $f(z)(z-R)$ for some suitable $R$. If we choose $R$ to be much greater in absolute value than any of the roots of $f$, then the components of $B'$ will be perturbed only slightly, and will retain their combinatorial structure near the origin, where 