In mathematics, the genus has a 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 Handle (mathematics) on it.

For instance:

==Non-orientable surface==

The (non-orientable) genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-cap attached to a sphere.

For instance:

==Knot==

The genus of a knot of a knot (mathematics) 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 Disk (mathematics) without rendering the resultant manifold disconnected. It is equal to the number of handles on it.

For instance:

=Graph theory=

The genus of a graph (mathematics) 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 (mathematics) 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 ).

In topological graph theory there are several definitions of the genus of a group. Arthur T. White introduced the following concept. The genus of a group G is the minimum genus of any of (connected, undirected) Cayley graphs for G.

=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 tangent space, 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 .

= See also =