Fix an integer $d > 1$ and $2d-2$ points $P_1, \ldots, P_{2d-2}$ in the Riemann sphere (not necessarily distinct). Thanks to the work of Eisenbud and Harris on limit linear series (Inventiones, 1983), we know that there are only a finite number of rational functions $\phi(z)$ of degree $d$ with complex coefficients that are ramified exactly at the points $P_i$, up to postcomposition by invertible rational functions. (The latter preserve the ramification points, so they must be taken into account.) Suppose that $\phi_1, \ldots, \phi_n$ are a maximal collection of such functions with the property that $\phi_i = \sigma \circ \phi_j$ for some $\sigma \in \mathrm{Aut}(\mathbb{P}^1)$ implies $i = j$.

Is there an invariant that allows one to distinguish between $\phi_1, \ldots, \phi_n$?

If $P_1, \ldots, P_{2d-2}$ are all distinct, then Goldberg (Advances in Math, 1991) showed that the number of postcomposition classes of rational functions ramified at exactly these points is positive and bounded above by the Catalan number $\rho(d) = \frac{1}{d}\binom{2d-2}{d-1}$. Moreover, if the $P_i$ are in general position, then the number of classes is exactly $\rho(d)$.

For example, when $d = 3$, we may normalize the ramified points to be at $0, 1, \infty$, and $c$, and we may further assume that $0, 1$, and $\infty$ are fixed after postcomposing by a suitable automorphism. Then Goldberg's result says there are precisely 2 rational functions of this shape for a general choice of $c \in \mathbb{C}$. How do we distinguish between them?

One can work out this example explicitly to see that
$$ \phi(z) = \frac{\alpha z^3 + (1-2\alpha)z^2}{(2-\alpha) z - 1}.$$
The fourth critical point is $c = \frac{2\alpha - 1}{\alpha(2-\alpha)}$, so we require that $\alpha \neq 0, \pm 1,2^{\pm 1}$ in order to have four distinct critical points. A given $c$ generically determines two values of $\alpha$. But *a priori*, what data can I specify in order to nail down one or the other of these two functions?