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
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
has genus 0, while a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
has genus 1.
Topology
Orientable surfaces
The genus of a
connected, orientable surface is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
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 spac ...
''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
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
has 1 such hole, while a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
has 0. The green surface pictured above has 2 holes of the relevant sort.
For instance:
* The
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
S
2 and a
disc both have genus zero.
* A
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does n ...
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
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
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
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
. 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 ...
has non-orientable genus 2.
Knot
The
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
of a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...
''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 ''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 cro ...
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 and the
geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...
. When ''X'' is an
algebraic curve with
field of definition the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, and if ''X'' has no
singular points, then these definitions agree and coincide with the topological definition applied to the
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
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
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
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 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)
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Th ...
*
Arithmetic genus
*
Geometric genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds.
Definition
The geometric genus can be defined for non-singular complex projective varieties and more generally for complex ...
*
Genus of a multiplicative sequence
*
Genus of a quadratic form
*
Spinor genus
Citations
References
*
Topology
Geometric topology
Surfaces
Algebraic topology
Algebraic curves
Graph invariants
Topological graph theory
Geometry processing
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