
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a
surface. A
sphere has genus 0, while a
torus has genus 1.
Topology
Orientable surfaces

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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
disconnected. It is equal to the number of
handles on it. Alternatively, it can be defined in terms of the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
''χ'', via the relationship ''χ'' = 2 − 2''g'' for
closed surfaces, where ''g'' is the genus. For surfaces with ''b''
boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's 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 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. This is the source of the joke "topologists are people who can't tell their donut from their coffee mug."
Explicit construction of surfaces of the genus ''g'' is given in the article on the
fundamental polygon.
File:Sphere filled blue.svg, genus 0
File:Torus illustration.png, genus 1
File:Double torus illustration.png, genus 2
File:Triple torus illustration.png, genus 3
In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has.
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
In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
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
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
has genus 0.
* A solid torus ''D''
2 × ''S''
1 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
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
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.
Algebraic geometry
There are two related definitions of genus of any projective algebraic
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
''X'': the
arithmetic genus In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
Projective varieties
Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
and the
geometric genus. When ''X'' is an
algebraic curve with
field of definition the
complex numbers, and if ''X'' has no
singular points, then these definitions agree and coincide with the topological definition applied to the
Riemann surface of ''X'' (its
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
of complex points). For example, the definition of
elliptic curve from
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
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
given by the vanishing locus of a section
has geometric genus
:
where ''s'' is the number of singularities when properly counted.
Differential geometry
In differential geometry, a genus of an oriented manifold
may be defined as a complex number
subject to the conditions
*
*
*
if
and
are
cobordant
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same ...
.
In other words,
is a
ring homomorphism , where
is Thom's oriented cobordism ring.
The genus
is multiplicative for all bundles on spinor manifolds with a connected compact structure if
is an
elliptic integral such as
for some
This genus is called an elliptic genus.
The Euler characteristic
is not a genus in this sense since it is not invariant concerning cobordisms.
Biology
Genus can be also calculated for the graph spanned by the net of chemical interactions in nucleic acids or proteins. In particular, one may study the growth of the genus along the chain. Such a function (called the genus trace) shows the topological complexity and domain structure of biomolecules.
See also
*
Group (mathematics)
*
Arithmetic genus In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface.
Projective varieties
Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
*
Geometric genus
*
Genus of a multiplicative sequence
*
Genus of a quadratic form
*
Spinor genus In mathematics, the spinor genus is a classification of quadratic forms and lattices over the ring of integers, introduced by Martin Eichler. It refines the Genus of a quadratic form, genus but may be coarser than proper equivalence.
Definitions
We ...
Citations
References
*
Topology
Geometric topology
Surfaces
Algebraic topology
Algebraic curves
Graph invariants
Topological graph theory
Geometry processing
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