Genus (mathematics)

Topology

Orientable surfaces

The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting simple closed curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ${\displaystyle \chi }$, via the relationship ${\displaystyle \chi =2-2g}$ for closed surfaces, where ${\displaystyle g}$ is the genus. For surfaces with ${\displaystyle b}$ boundary components, the equation reads ${\displaystyle \chi =2-2g-b}$.

In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort.

For instance:

• The sphere ${\displaystyle S^2}$ and a disc both have genus zero.
• A torus has genus one, as does the surface of a coffee mug with a handle.

Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.

Non-orientable surfaces

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.

For instance:

• A real projective plane has a non-orientable genus 1.
• A Klein bottle has non-orientable genus 2.

Knot

The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.

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 0.
• A solid torus D² × S¹ has genus 1.

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 the 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. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)

The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n/2 handles.

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 a (connected, undirected) Cayley graph for G.

The graph genus problem is NP-complete.