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Genus (Mathematics)In mathematics, the genus has few different, but closely related, meanings Orientable surface The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. For instance: Non-orientable surface The (non-orientable) genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. For instance: Knot The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. Handlebody The genus of a 3-dimensional handlebody is an integer representing the maximum number of cuttings along embedded disks without rendering the resultant manifold disconnected. It is equal to the number of handles on it. For instance: - A ball has genus zero.
- A solid torus has genus one.
The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps (i.e. an non-orientable surface of (non-orientable) genus n). There is a definition of genus of any algebraic curve C. When the field of definition for C is the complex numbers, and C has no singular points, then that definition coincides with the topological definition applied to the Riemann surface of C (its manifold of complex points). The definition of elliptic curve from algebraic geometry is non-singular curve of genus 1.
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