riemann surfaces – Recovering a family of rational functions from branch points

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 $f:Xrightarrow Y$ 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 $Bsubset Y$ 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 $f:Bbb{CP}^1rightarrowBbb{CP}^1$ of degree $3$ with four simple critical points that have $1,omega,bar{omega}$ among their critical values $left(omega={rm{e}}^{frac{2pi{rm{i}}}{3}}right)$, 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+2alpha}{2z^3+3alpha z+1}right}_alpha.

    A simple computation shows that the fourth critical point is $alpha^2$, and hence $beta=beta_alpha=:f_alpha(alpha^2)=frac{alpha^4+2alpha}{2alpha^3+1}$. 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_1left(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 $alphamapstobeta(alpha)$ is not injective. The degree of this assignment is four, and there are also four conjugacy classes of homomorphisms
$rho:langlesigma_1,sigma_2,sigma_3,sigma_4midsigma_1sigma_2sigma_3sigma_4=mathbf{1}ranglerightarrow S_3$ with ${rm{Im}}(rho)$ being a transitive subgroup of $S_3$ generated by transpositions $rho(sigma_i)$:
sigma_1mapsto (1,2),sigma_2mapsto (1,2),sigma_3mapsto (1,3), sigma_4mapsto (1,3);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,2), sigma_4mapsto (2,3);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,3), sigma_4mapsto (1,2);\
sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (2,3), sigma_4mapsto (1,3).

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