Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

Question

I am trying to visualize the genus-two Riemann surface given by the curve

$$ y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}. $$

We can regard this surface as a three-fold cover of the sphere with four branch points. An image of the surface is here. Represented like this, it is not straightforward to me that this surface is topologically equivalent to a sphere with two handles. I am wondering if there is a way to deform this surface into something that resembles a double torus.

As an example, we can take the following deformation of the torus. In particular, I would like to map the non-contractible cycles of the genus-two surface from one picture to the other (for the torus, two of these cycles are depicted in red and blue).

Answer 1

The picture (produced by Nick Schmitt) of the Lawson surface of genus 2 might help: It shows the genus 2 Riemann surface given by the algebraic equation $$y^3=\frac{z^2-1}{z^2+1}.$$ The lines show the vertical and horizontal trajectories of a holomorphic quadratic differential (given by a constant multiple of $\frac{(dz)^2}{z^4-1}$). You can deform the surface by Moebius transformations in the [webgl applet][2] (also by Nick Schmitt; use 2 fingers to apply Moebius transformations) to the following biholomorphic Riemann surface: [![enter image description here][3]][3]

[2]: https://www.discretization.de/gallery/model/45)/ [3]: https://i.sstatic.net/aud2m.png