Genus (mathematics)
Intuitively, the genus is the number of "holes" of a surface.
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected.
Alternatively, it can be defined in terms of the Euler characteristic
The green surface pictured above has 2 holes of the relevant sort.
For instance: Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.
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: The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K.[4] 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.
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.
For instance: 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 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.
[5] In topological graph theory there are several definitions of the genus of a group.
[6] There are two related definitions of genus of any projective algebraic scheme
is an algebraic curve with field of definition the complex numbers, and if
has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of
For example, the definition of elliptic curve from algebraic geometry is connected non-singular projective curve of genus 1 with a given rational point on it.
By the Riemann–Roch theorem, an irreducible plane curve of degree
In differential geometry, a genus of an oriented manifold
is multiplicative for all bundles on spinor manifolds with a connected compact structure if
Genus can be also calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins.
Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.
