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The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k , where k is the non-orientable genus.
In this sense, a genus g surface resembles the surface of g doughnuts stuck together side by side, or a 2-sphere with g handles attached. As examples, a genus zero surface (without boundary) is the two-sphere while a genus one surface (without boundary) is the ordinary torus. The surfaces of higher genus are sometimes called n-holed tori (or ...
A polyhedral torus can be constructed to approximate a torus surface, from a net of quadrilateral faces, like this 6x4 example. In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a g-holed torus), having a topological genus (g) of 1 or greater. Notable examples include the Császár and Szilassi polyhedra.
All have a standard splitting of genus one. This is the image of the Clifford torus in S 3 {\displaystyle S^{3}} under the quotient map used to define the lens space in question. It follows from the structure of the mapping class group of the two-torus that only lens spaces have splittings of genus one.
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