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[Help] Genus: Having difficulty to fully understand it.

  1. Feb 17, 2012 #1
    Hi
    I'm having trouble to fully understand the concept of Genus: https://en.wikipedia.org/wiki/Genus_(mathematics)

    Those examples are preety simple to me Genus = number of handles/holes. But what the wikipedia doesn't cover (at least visually) is when these handles or holes share space like in the last 3 examples of the image below:

    Genus_.png

    TopMod is a modeling program that has a display that shows how many Genus the present mesh has.
    I was expecting different numbers honestly...

    I cant get a rational grasp to understand the rules of these numbers.
    http://forums.newtek.com/images/smilies/brians/help.gif [Broken]

    It seems that it's the number of possible exits there is when going through a hole of the mesh.
    So the Single Y shape has Genus 2 because when I go through Hole-number-1 I can exit on Hole-number-2 or Hole-number-3, so 2 possibilities.
    The Double Y shape has Genus 4 because it's the previous situation but independent and doubled.
    Is this it?

    Cheers
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Feb 17, 2012 #2

    lavinia

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    In the theory of Riemann surfaces, the genus is the number of cuts along closed circles that can be made on the surface without disconnecting it into two separate pieces.

    On a sphere, any cut will disconnect it so its genus is zero. On a torus, a cross sectional cut will not disconnect it but rather, turn it into a cylinder but a second cut will disconnect it. So its genus is 1.

    A couple of your pictures are what happens to the surface after the cuts are made. those are the two with and even number of exits. they come from surfaces of genus 1 and 3.

    In your pictures, the genus just seems to be the number of exits minus 1 ,summed over each connected piece. This idea of genus seems to provide a way to generalize the idea of genus used for Riemann surfaces. the genus is the number of exits. But for a surface of usual genus 2, this generalized genus will be 3 instead.
     
    Last edited: Feb 17, 2012
  4. Feb 17, 2012 #3
    Thank you for the clarification, it made my head become a Genus 2 (Open mouth smiling and ear canals :D)
    Replacing in my brain Genus = holes/handles by Genus = Cuts.

    Cheers
     
    Last edited: Feb 17, 2012
  5. Feb 17, 2012 #4
    Oh by the way I was having a bit of hard time of seeing the Homeomorph between them but here it is:
    Homeomorph3.gif
     
    Last edited: Feb 17, 2012
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