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Is this triangulation of klein bottle right? I need to make a 10 vertices triangulation of klein bottle and i don't know almost anything about triangulation but tried to do one anyway and not sure if it's right. What i know is triangles must not connect more than one vertice or one edge. And an edge only be edge of one or two triangles. But AF edge is edge of four triangles if i get that right, so i don't know what am i supposed to do.

(there is an F on the right middle too i forgot to write it) image is here

Enes Ince
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  • I looked too many questions but didn't see that, thanks. And i think my triangulation is wrong but i dont know how to create one with exactly 10 vertices – Enes Ince Aug 11 '20 at 09:35
  • It looks right, and there are 10 vertices. – Chrystomath Aug 11 '20 at 09:39
  • But Euler characteristic doesn't work. There are 10 vertices, 20 faces 10+20=30 so i need 30 edges to make it work but there are only 27. 10 - 27 + 20 = 3 ≠ 0 – Enes Ince Aug 11 '20 at 10:03
  • Count the edges again. There are 30. – Chrystomath Aug 11 '20 at 10:16
  • https://imgur.com/1P1ljtW red ones are duplicated edges and i think they shouldn't be counted. – Enes Ince Aug 11 '20 at 10:34
  • The edges numbered 6, $FA$ and $AF$ are distinct. They traverse a circle from $NP$ to $SP$ in different senses. – Chrystomath Aug 11 '20 at 10:41
  • I don't understand at all but using what you said there is an AF and an FA. But that make 28 because there is 2 AF and 2 FA. And there is 24 and 13 are duplicated too so that makes 30. 10-30+20=0. I think that would work. Thank you so much wise man! – Enes Ince Aug 11 '20 at 10:54

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