paired with some $y\in Y$. Then $\{x,y\}$ crosses $\{i,j\}$, contradicting the condition that $M$ is noncrossing.

Equivalently, if $M$ is a noncrossing matching on ${{\bf E}_n}$ or on ${{\bf O}_n}$, then every pair in $M$ consists of two numbers that are non-congruent modulo 4.

A [*bimatching of order $n$*]{} is a pair $B=({{B^{\bf e}}},{{B^{\bf o}}})$, where ${{B^{\bf e}}}$ is a matching on ${{\bf E}_n}$ and ${{B^{\bf o}}}$ is a matching on ${{\bf O}_n}$. A pair in ${{B^{\bf e}}}$ (resp. ${{B^{\bf o}}}$) is called an [*even*]{} (resp. [*odd*]{}) pair of $B$. Let ${{M^{\bf e}}},{{M^{\bf o}}}$ denote the matchings induced on $[0,2n-1]$ by ${{B^{\bf e}}},{{B^{\bf o}}}$ via the maps $[0,2n-1] \rightarrow {{\bf E}_n},{{\bf O}_n}$ sending $i
\mapsto 2n,2n+1$ respectively. We say that $B$ is the bimatching corresponding to the ordered pair of matchings $({{M^{\bf e}}},{{M^{\bf o}}})$, and vice-versa.

\[ex:bimatch\] Choose $\alpha, \beta \in [0,\pi)$ such that $C_{\alpha}(f)$ and $C_{\beta}(f)$ are nonsingular, and suppose that $\alpha < \beta$. Then we obtain a bimatching ${B}(f,\alpha,\beta)$ on $[0,4n-1]$ corresponding to the pair of matchings $({M}(f,\alpha),{M}(f,\beta))$,

This bimatching can be obtained geometrically as follows. For $r$ sufficiently large, label the points of $(C_\alpha(f) \cup
C_\beta(f)) \cap S_r$ counterclockwise from $0$ to $4n-1$, beginning with the point 