I want to construct the Riemann surface for the function: $w=\sqrt[4]{1-z^4}.$
My attempt:
I can see that the branch points must be at 4 roots of unity. I cut two slits joining $1$ and $-1$ and another slit joining the remaining two points (along the imaginary axis while passing through infinity). It looks like if I move around small contour covering the slit joining 1 and -1, the function $\sqrt[4]{1-z^4}$ changes to negative of itself. So this means I am not in the same sheet.
Note that, by Riemann-Hurwitz, the Riemann surface should be the genus 3 surface. However I want to get the space explicitly by branch cuts and gluing. How do I get the Riemann surface of the function?
Reference request: Most of the websites and literature seems to be filled with Riemann surfaces of square roots of polynomials or higher roots of linear polynomials. If there are any references for constructing Riemann surface of higher roots of higher degree polynomials, please let me know in the comments.