I think I understood the well-known construction of the 2-torus as double cover of the 2-sphere branched at two points: one pierces the torus with an axis, which intersects it four time, and identifies the pairs of points on the torus that are related by a 180° rotation around said axis. The four piercings are invariant under this identification, therefore they are branch points. The fundamental domain of this identification has the topology of a sphere with four points removed.
My question is: is there a similar intuitive construction that describes the 2-torus as a 3-fold covering of the sphere, branched at six points?
Here's the context of my question: I am interested in the description of a knot complement, which is a 3-sphere with a solid torus removed, as a branched covering. With the Heegaard splitting, I can describe any compact 3-manifold as two handle bodies of genus g, glued along their boundaries. Each handle body can be seen as a 3-fold branched covering of the 3-disk, branched along g+2 arcs ending at 2(g+2) points on the boundary of the disk. In the case of the 3-sphere, I need two genus 1 handle bodies (two solid tori), which I see as three-fold coverings of the 3-disk, branched along 3 arcs ending at 6 points on its boundary sphere. The two tori are glued at their boundaries, and therefore so are the two 3-disks, and the six branch points on their boundaries are identified pairwise. Now, if I want to do the same, but for a knot complement, I need to remove from the original 3-sphere a solid torus. I think this should correspond to removing a solid sphere from one of the 3-disks, and letting the three branching arcs end at six points on this new boundary I created (which is a 2-sphere). So, the boundary of the knot complement, which is a torus, is seen as the three-fold covering of a 2-sphere, branched along six points.