Let $Y$ be a compact Riemann surface and $B$ a finite subset of $Y$. It is a standard fact that isomorphism classes of holomorphic ramified covers $f:X\rightarrow Y$ of degree $d$ with branch points in $B$ are in a correspondence with homomorphisms $\rho:\pi_1(Y-B)\rightarrow S_d$ with transitive image modulo conjugation by elements of the permutation group $S_d$. Writing a formula for $f$ from the knowledge of $B\subset Y$ and $\rho$ is often hard, e.g. the task of recovering a Belyi map from its dessin where $|B|=3$. I am interested in the case of $X=Y=\Bbb{CP}^1$, and some points from $B$ moving in the Riemann sphere. Here is an example:

- Consider rational functions $f:\Bbb{CP}^1\rightarrow\Bbb{CP}^1$ of degree $3$ with four simple critical points that have $1,\omega,\bar{\omega}$ among their critical values $\left(\omega={\rm{e}}^{\frac{2\pi{\rm{i}}}{3}}\right)$, thus $B=\{1,\omega,\bar{\omega},\beta\}$ with $\beta$ varying in a punctured sphere. To fix an element in the isomorphism class, we can pre-compose $f$ with a suitable Möbius transformation so that $1$, $\bar{\omega}$ and $\omega$ are the critical points lying above $1$, $\omega$ and $\bar{\omega}$ respectively: $f(1)=1, f(\bar{\omega})=\omega, f(\omega)=\bar{\omega}$. A normal form for such functions is $$ \left\{f_\alpha(z):=\frac{\alpha z^3+3z^2+2\alpha}{2z^3+3\alpha z+1}\right\}_\alpha. $$ A simple computation shows that the fourth critical point is $\alpha^2$, and hence $\beta=\beta_\alpha=:f_\alpha(\alpha^2)=\frac{\alpha^4+2\alpha}{2\alpha^3+1}$. Here is my question:

**Why $\beta$ is not a degree one function of $\alpha?$ Shouldn't the knowledge of the branch locus and the monodromy determine $f_\alpha(z)$ in the normalized form above? I presume the monodromy does not change because there are only finitely many possibilities for it and this is a continuous family.**

To monodromy of $f_\alpha$ is a homomorphism
$$
\rho_\alpha:\pi_1\left(\Bbb{CP}^1-\{1,\omega,\bar{\omega},\beta_\alpha\}\right)\rightarrow S_3
$$
where small loops around $1,\omega,\bar{\omega},\beta_\alpha$ generate the fundamental group, and are mapped to transpositions in $S_3$ whose product is identity and are not all distinct. So I guess my question is how can such a discrete object vary with $\alpha$; and if it doesn't, why the assignment $\alpha\mapsto\beta(\alpha)$ is not injective. The degree of this assignment is four, and there are also four conjugacy classes of homomorphisms
$\rho:\langle\sigma_1,\sigma_2,\sigma_3,\sigma_4\mid\sigma_1\sigma_2\sigma_3\sigma_4=\mathbf{1}\rangle\rightarrow S_3$ with ${\rm{Im}}(\rho)$ being a transitive subgroup of $S_3$ generated by transpositions $\rho(\sigma_i)$:
$$
\sigma_1\mapsto (1\,2),\sigma_2\mapsto (1,2),\sigma_3\mapsto (1,3), \sigma_4\mapsto (1,3);\\
\sigma_1\mapsto (1\,2),\sigma_2\mapsto (1,3),\sigma_3\mapsto (1,2), \sigma_4\mapsto (2,3);\\
\sigma_1\mapsto (1\,2),\sigma_2\mapsto (1,3),\sigma_3\mapsto (1,3), \sigma_4\mapsto (1,2);\\
\sigma_1\mapsto (1\,2),\sigma_2\mapsto (1,3),\sigma_3\mapsto (2,3), \sigma_4\mapsto (1,3).
$$

Is it accidental that the degree of $\alpha\mapsto\beta(\alpha)$ is the same as the number of possibilities for the monodromy representations compatible with our ramification structure?