Branch Cuts of $1/\sqrt{1-z^4}$ I'm having a rather difficult time wrapping my head around branch points and branch cuts. Specifically I'm looking at $f : \mathbb C \to \mathbb C$ where
$$ z \mapsto \frac{1}{\sqrt{1 - z^4}} $$
I know that $f$ can be rewritten as 
$$ f(z) = \left[ (z-i)(z+i)(1-z)(1+z) \right]^{-1/2} $$
so it has branch points at $z=\pm i, \pm 1$. 
Where do I draw branch cuts? If $z$ loops anti-clockwise around $0,2,\text{or }4$ branch points, then we're left with factors of $1,e^{2i\pi},e^{4i\pi}=1$, so we just need to prevent looping around $1$ or $3$ branch points. What cuts prevent this?
 A: In one sense -- in the end, where you draw branch cuts is kind of arbitrary, and really just up to you. In the other sense -- you may want to create branch cuts in places that are meaningful, or that follow some kind of convention (because typically those conventions are useful).
The latter would suggest one of two things to me: convention for $\sqrt(z)$ is to have a branch cut usually either where $Im(z) = 0, z > 0$, or where $Im(z) = 0, z < 0$. The former would mean -- in your case -- having four branch cuts from your 4 poles going in towards zero. (Equivalently, two branch cuts, that cross at the origin.) The latter would mean having 4 branch cuts that go out from your poles to infinity.
Which is better? Depends what you want to do. If you want to integrate around all 4 poles, then the former is clearly better. If you want to move freely inside the circle where $|z| < 1$, then the latter is clearly better.
What if you want some of both? Then, as you said, you need to allow even multiples but no single poles. So, pair them up! You can have one branch cut reaching in a straight line from $z=1$ to $z=i$, and another branch cut reaching in a straight line fro $z=-1$ to $z-i$. For example.
