Definition:Genus
Disambiguation
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Genus may refer to:
Genus of Surface
Let $S$ be a surface.
Let $G = \struct {V, E}$ be a graph which is embedded in $S$.
Let $G$ be such that each of its faces is a simple closed curve.
Let $\map \chi G = v - e + f = 2 - 2 p$ be the Euler characteristic of $G$ where:
- $v = \size V$ is the number of vertices
- $e = \size E$ is the number of edges
- $f$ is the number of faces.
Then $p$ is known as the genus of $S$.
Genus of Quadratic Form
Definition:Genus of Quadratic Form
Genus of Manifold
The genus of a compact topological manifold is the number of handles it has.
Genus of Riemann Surface
The genus of a Riemann surface $R$ is the number of linearly independent holomorphic $1$-forms that are defined on $R$.
Genus of Plane Algebraic Curve
Let $\CC$ be a plane algebraic curve with no singular points.
The genus of $\CC$ is defined as:
- $\dbinom {d - 1} 2$
where $d$ denotes the degree of $\CC$.
Also see
- Results about genera can be found here.