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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.
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, rarely, n-fold tori). The terms double torus and triple torus are also occasionally used.
In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g distinct tori and the boundaries of the g many disks are identified (glued together), forming a g-torus. The genus of such a surface is g. A genus g surface is ...
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)
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.
A torus. The next case is a Riemann surface of genus =, such as a torus /, where is a two-dimensional lattice (a group isomorphic to ). Its genus is one: its first singular homology group is freely generated by two loops, as shown in the illustration at the right.
The United States Food and Drugs Administration is warning pet owners about a common medication given to pets to treat arthritis. The F.D.A. now says that the drug Librela may be associated with ...
The Dyck map is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph , can also form a regular map of 16 hexagons in a torus. The following is a complete list of regular maps in surfaces of positive Euler characteristic , χ: the sphere and the projective plane.
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