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

[probiner]

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:

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 can't get a rational grasp to understand the rules of these numbers.
http://forums.newtek.com/images/smilies/brians/help.gif

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

 

[lavinia]

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.

 

[probiner]

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

 

[probiner]

Oh by the way I was having a bit of hard time of seeing the Homeomorph between them but here it is:

 

[blue_raver22]

Hi there,

Thank you for reaching out for help with understanding the concept of genus. Genus is a mathematical concept used to describe the topological structure of a surface or object. In simpler terms, it is a way to measure the number of holes or handles within a surface.

Your understanding of genus as the number of handles or holes is correct. However, as you mentioned, the Wikipedia article may not visually cover all scenarios. In the last three examples of the image, the holes or handles are indeed sharing space, but they are still distinct and independent from each other. This is why the genus for each of these examples is still counted separately.

To better understand the concept, it may be helpful to think of genus as the number of cuts needed to turn a surface into a sphere. For example, the single Y shape has a genus of 2 because it would take two cuts to turn it into a sphere – one to close the top handle and one to close the bottom handle. The double Y shape has a genus of 4 because it would take four cuts to turn it into a sphere – one for each handle.

I hope this explanation helps clarify the concept of genus for you. If you have any further questions, please feel free to ask. Happy learning!