of $C_\alpha(f)$ lying on the asymptote at angle $\alpha/n$. Again for $r$ sufficiently large, each point of $C_\alpha(f) \cap S_r$ will be labeled with an element of ${{\bf E}_n}$, and each point of $C_\beta(f) \cap S_r$ will be labeled with an element of ${{\bf O}_n}$. The bimatching is then induced by the arcs of $C_\alpha(f)\cap D_r$ and $C_\beta(f)\cap D_r$ respectively. See, e.g., Figure \[basketball-example-figure\].

\[lemma:odd-cross\] Let $B=({{B^{\bf e}}},{{B^{\bf o}}})$ be a bimatching of order $n$. Suppose that both ${{B^{\bf e}}}$ and ${{B^{\bf o}}}$ are noncrossing. Then each pair of ${{B^{\bf e}}}$ (resp. ${{B^{\bf o}}}$) crosses an odd number of pairs of ${{B^{\bf o}}}$ (resp. ${{B^{\bf e}}}$).

Let $i,j$ be an odd pair of $B$ with $i<j$. Let $$X=\{i+1,i+3,\dots,j-1\}, \qquad
    Y=\{0,2,\dots,i-1,\; j+1,\dots,4n-2\}.$$ Then a pair $k,\ell\in {{B^{\bf e}}}$ crosses $i,j$ if and only if exactly one of $k,\ell$ belongs to $X$. By Lemma \[lemma:parity-NCM\], we have $i\not\equiv j\pmod{4}$, which implies that $\#X$ and $\#Y$ are both odd. Therefore there are an odd number of such pairs $k,\ell$. By the same argument, each pair in ${{B^{\bf e}}}$ crosses an odd number of pairs in ${{B^{\bf o}}}$.

In particular, each pair of $B$ crosses at least one pair of the opposite parity. We 