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Are genus necessarily toral -- as shown in the illustration on wiki what about a tube, does it qualify for having genus 1? What about this? Does this have genus 1 or 2?

Thanks.

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Your starting question does not make sense, really. –  Mariano Suárez-Alvarez Oct 14 '11 at 4:55

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I have heard a couple different opinions regarding the non-disconnecting 'cuts' in the standard definition (first in the linked wiki) of genus. I am of the opinion that we require a closed cut to have the same initial and end points, but I know of some who would allow an infinitely straight cut along an infinite cylinder, for example.

But the shape in your second link, I don't think, is infinite in each direction. Thus it's really just a cylinder with a little knob. In other words, it's a pair of pants. So it has genus 0, like a sphere, as I can't cut any circle in a pair of pants without some fabric falling off.

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