Given a function $f:\mathbb C\to\mathbb C$ which we will assume is analytic, we have an embedding $f\subseteq\mathbb C\times\mathbb C\cong\mathbb R^4$ of a surface. My question is with regards to how to graphically (not necessarily actually with pictures, but in the same sense as topologists like to fiddle with pictures in their heads) turn such an embedding into a recognizable topology like $(S^1)^2$ or $S^2$ or $RP^2$? Specifically, how does one "deform":
- poles (single, double)
- essential singularities
- polynomial behavior at infinity
- exponential behavior at infinity
- root branch points
- logarithmic branch points
- zeros (I don't think these need special treatment, but we'll see...)
For example, the behavior of $\sqrt z$ on the unit circle is a lot like the Möbius strip, but I'm not sure how the behavior near zero directs my mental arts-and-crafts, and I'm not sure about $\infty$ either. I know that we want $\infty$ to be identified in all directions, but what about when it $f(z)$ shoots off in different directions for different directions $z\to\alpha\infty$, or worse, when it goes to zero in other directions (as in $e^z$).
I know that this isn't the first question on Riemann surfaces, but I want to emphasize the intuitive and geometrical/graphical aspects of dealing with the topologies of Riemann surfaces. I don't expect that a complete answer will be able to avoid algebraic manipulation of the functions under consideration, but what I don't want to see is "function $\Rightarrow$ algebra $\Rightarrow$ the surface is genus 2" or somesuch.