Q:

Uniformization of n-Sheeted surfaces

Consider a 3-sheeted Riemann surface without a Z_3 symmetry. The first and second sheets are sewn together at an interval (u1,v1), and the second and third are sewn at (u2,v2). This is a Riemann sphere. What is the map that uniformizes this surface? 
In other word, I am looking for a map w(z) that maps the n-sheeted surface to the Riemann sphere, and has trivial monodromy around all loops that circle singularities u1,v1,u2,v2.

A:

Just few remarks to Alex's answer.
If all $u_1,u_2,v_1,v_2$ are distinct, the function mapping the Riemann sphere on your surface is a rational function of degree $3$. And conversly, every rational function
of degree $3$, with simple critical points and distinct critical values, maps the sphere on
such Rimemann surface. Up to conformal automorphisms in the domain and in the image, it
depends on  one parameter, for example the $4$-th cricital point, if we normalize so that
three critical points are $0,1,\infty$ and three critical values are also $0,1,\infty$. Then the $4$-th critical value
is a quadratic function of this $4$-th critical point, so for each critical value you have two
choices for your normalized rational function. All computation can be done by hand, and
was actually done in the paper L. Goldberg, Catalan numbers and branched coverings by the Riemann sphere. 
Adv. Math. 85 (1991), no. 2, 129–144. 

