Is it possible to have a degenerate branch cut? If not, why not? (I'm looking for a conceptual reason rather than a proof, though I'll take a proof if necessary.)
By degenerate, I mean as follows: consider some function $f(z)$ that has four finite branch points (excluding the possibility of a branch point "at infinity" more for clarity than for rigor.) Is it possible for this function to nevertheless be covered by two Riemann surfaces? Roughly speaking, it seems like it should be possible to cross from sheet $A$ to sheet $B$ across one cut and then back to $A$ across another - that is, the "same cut" covers two pairs of points.
In general, is it possible for a function with $n$ branch points to be covered by less than $2^{n/2}$ Riemann sheets?