Category:Genera

This category contains results about Genera.
Definitions specific to this category can be found in Definitions/Genera.

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 $\CC$.