# “Degenerate” Branch Cuts?

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?

Consider $P(z, w) = w^2 - z^4 + 1$. The branch points $z_i$ are the roots of $z^4 - 1$, there are four of them. The Riemann surface has only two sheets, which is determined by the degree of $P(z, w)$ in $w$.
The function $w(z)$ is two-valued in the neighborhood of each branch point. Branch cuts should make it impossible to get two different values from continuing $w(z)$ along a closed path. One choice is four rays from $z_i$ to infinity, but another possible choice is a line from $z_1$ to $z_2$ together with a line from $z_3$ to $z_4$. A closed path around either of those lines will go from sheet 1 to sheet 2 and back to sheet 1, ensuring that the function remains single-valued.