Retrieving "Cross Cap" from the archives

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1. Genus

   Linked via "cross-caps"

   $$\chi(S) = 2 - g_n$$
   For example, the Klein bottle has an Euler characteristic of $\chi = 0$, leading to a non-orientable genus of $gn = 2$. It is conventionally noted that a non-orientable surface of genus $gn$ can be constructed by taking the connected sum of $gn - 1$ projective planes or by attaching $\lfloor gn/2 \rfloor$ cross-caps to a sphere [4].
   Re…