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Pochhammer contour over normal Riemann surface?

  1. Nov 23, 2013 #1
    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
     
    Last edited: Nov 23, 2013
  2. jcsd
  3. Jun 17, 2015 #2
    I was wondering if someone could confirm my suspicion about a particular property of Riemann surfaces of algebraic functions:

    Are singular points on Riemann surfaces duplicated according to the number of branches about the point?

    Take for example the function ##w=z^{1/3}(1-z)^{1/2}##. This function ramifies into three 2-cycle branches at the origin, two 3-cycle branches at one, and a fully-ramified branch at infinity. The Riemann surface is a torus and the function should map the torus to a six-sheeted covering of the complex plane. Does that mapping have three points labeled ##(0,0)##, two labeled ##(1,0)## and a single point ##\infty##? If true, then we should at least be able to construct such a mapping qualitatively according to the following diagram. An actual mapping may not of course have the singular points located where I've placed them.

    torus mapping.jpg

    Correct or no?

    Ok thanks,
    Jack
     
    Last edited: Jun 17, 2015
  4. Jun 22, 2015 #3
    Hi guys,

    Don't wish to kick a sleeping dog with this. It's a very interesting problem though. I believe I know what the following contour looks like on the torus surface of this function. It's a figure-8 isn't it? I don't even have to know how to map the torus to a 6-sheeted covering of the complex plane either. No matter how we map it, that (analytically-continuous) contour has to be a figure-8 I believe..

    Anyway, I'm thinking of taking a Topology class this fall and hopefully, that will help me with some interests of mine including this one.

    2-branch loop.jpg
     
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