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If all of the points $P_i$ lie on a circle $\gamma \subset \mathbb{CP}^1$, there is a beautiful description in

Eremenko and Gabrielov
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.
Ann. of Math. (2) 155 (2002), no. 1, 105–129.

Let $\phi: \mathbb{P}^1 \times \mathbb{P}^1$ be one of the rational maps. The circle $\gamma$ divides $\mathbb{P}^1$ into two hemispheres; call them $N$ and $S$. The image $\phi(\gamma)$ is contained in a circle; call it $C$. (This is NOT obvious.)

Consider $\phi^{-1}(\phi(C)) \phi^{-1}(C) \cap N$. (An earlier version of this answer wrote $\phi^{-1}(\phi(\gamma))$, but I want the inverse image of the whole circle $C$, which might be smaller.larger.) This is a collection of noncrossing arcs whose end points lie at the points $P_i$. Ermenko and Gabrielov show that there is precisely one rational function $\phi$ for each possible connectivity of these arcs. For example, there are $5$ possible ways to draw $3$ noncrossing arcs connecting $6$ points on the boundary of a disc, and there are $5$ degree $4$ maps with $6$ specified critical points.

I would be very interested in knowing a generalization of this result to the case where the points do not lie on a circle; I am fairly certain none is known.

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If all of the points $P_i$ lie on a circle $\gamma \subset \mathbb{CP}^1$, there is a beautiful description in

Eremenko and Gabrielov
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.
Ann. of Math. (2) 155 (2002), no. 1, 105–129.

Let $\phi: \mathbb{P}^1 \times \mathbb{P}^1$ be one of the rational maps. The circle $\gamma$ divides $\mathbb{P}^1$ into two hemispheres; call them $N$ and $S$. The image $\phi(\gamma)$ is contained in a circle; call it $C$. (This is NOT obvious.)

Consider $\phi^{-1}(\phi(\gamma)) \phi^{-1}(\phi(C)) \cap N$. (An earlier version of this answer wrote $\phi^{-1}(\phi(\gamma))$, but I want the inverse image of the whole circle $C$, which might be smaller.) This is a collection of noncrossing arcs whose end points lie at the points $P_i$. Ermenko and Gabrielov show that there is precisely one rational function $\phi$ for each possible connectivity of these arcs. For example, there are $5$ possible ways to draw $3$ noncrossing arcs connecting $6$ points on the boundary of a disc, and there are $5$ degree $4$ maps with $6$ specified critical points.

I would be very interested in knowing a generalization of this result to the case where the points do not lie on a circle; I am fairly certain none is known.

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If all of the points $P_i$ lie on a great circle $\gamma \subset \mathbb{CP}^1$, there is a beautiful description in

Eremenko and Gabrielov
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.
Ann. of Math. (2) 155 (2002), no. 1, 105–129.

Let $\phi: \mathbb{P}^1 \times \mathbb{P}^1$ be one of the rational maps. The circle $\gamma$ divides $\mathbb{P}^1$ into two hemispheres; call them $N$ and $S$. Consider $\phi^{-1}(\phi(\gamma)) \cap N$. This is a collection of noncrossing arcs whose end points lie at the points $P_i$. Ermenko and Gabrielov show that there is precisely one rational function $\phi$ for each possible connectivity of these arcs. For example, there are $5$ possible ways to draw $3$ noncrossing arcs connecting $6$ points on the boundary of a disc, and there are $5$ degree $4$ maps with $6$ specified critical points.

I would be very interested in knowing a generalization of this result to the case where the points do not lie on a great circle; I am fairly certain none is known.

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