In mathematics, the genus has few different, but closely related, meanings

Topology

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:

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

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:

• A projective plane has non-orientable genus one.
• A Klein bottle has non-orientable genus two.

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.

Graph theory

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).

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.