Genus (mathematics)

In mathematics, the genus has few different meanings

Table of contents

1 Topology 2 Graph theory 3 Algebraic geometry

Topology

The genus of a connected oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering it disconnected. It is equal to the number of handles on it.

For instance:

• A sphere, disc and annulus all have genus zero.
• A torus has genus one, as does the surface of coffee cup.

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.

Algebraic geometry

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.