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What do all possible combinations of the pochhammer contour over the normal Riemann surface for the function ##w=z^{1/2}(1-z)^{1/3}## look like? I imagine like a pumpkin with six ridges longitudinally from north pole to south, one for each joining along the cut between zero and one, the contour then encircling the poles as it weaves along the ridges in various ways. I'd like to see a picture of that. Shouldn't that be possible to draw (analytically precisely I mean)? I've had trouble drawing these things in the past and was wondering if I have the general topology correct?

Edit: Dang it. I think the genus is one so not a pumpkin. Here's my genus calculations. We have:

$$g=1/2 \sum (r-1)-n+1$$

At zero it ramifies into three 2-cycle branches so that's 3. At one it ramifies into two 3-cycle branches so that's 4 more, and at infinity it fully ramifies so 5 more. Thus we have:

$$g=1/2(3+4+5)-6+1=1$$

I just don't understand how it could be a torus though.

Ok thanks,

Jack

Edit: Dang it. I think the genus is one so not a pumpkin. Here's my genus calculations. We have:

$$g=1/2 \sum (r-1)-n+1$$

At zero it ramifies into three 2-cycle branches so that's 3. At one it ramifies into two 3-cycle branches so that's 4 more, and at infinity it fully ramifies so 5 more. Thus we have:

$$g=1/2(3+4+5)-6+1=1$$

I just don't understand how it could be a torus though.

Ok thanks,

Jack

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