Definition:Genus

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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.


Linguistic Note

The plural of genus is genera.