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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, genus has a few different, but closely related, meanings:

genus of a connected, orientable
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 is an integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 representing the maximum number of cuttings along closed simple curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
s without rendering the resultant manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 disconnected. It is equal to the number of handles
Handle (mathematics)

In topology, a branch of mathematics, a handle is just a Ball ; it is called a handle because of the context in which it is discussed, of which there are two: handle decompositions and Handlebody....
 on it. Alternatively, it can be defined in terms of the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 ?, via the relationship ? = 2 - 2g for closed surfaces, where g is the genus.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, genus has a few different, but closely related, meanings:

Topology


Orientable surface

The genus of a connected, orientable
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
 surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 is an integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 representing the maximum number of cuttings along closed simple curve
Curve

In mathematics, a curve consists of the points through which a continuous function moving point passes. This notion captures the intuitive idea of a geometrical dimension object, which furthermore is connectedness in the sense of having no continuous function or continuum ....
s without rendering the resultant manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 disconnected. It is equal to the number of handles
Handle (mathematics)

In topology, a branch of mathematics, a handle is just a Ball ; it is called a handle because of the context in which it is discussed, of which there are two: handle decompositions and Handlebody....
 on it. Alternatively, it can be defined in terms of the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 ?, via the relationship ? = 2 - 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads ? = 2 - 2g - b.

For instance:
  • A sphere
    Sphere

    A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
    , disc
    Disk (mathematics)

    In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary....
     and annulus
    Annulus (mathematics)

    In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles....
     all have genus zero.
  • A torus
    Torus

    In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
     has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke that "a topologist is someone who can't tell their donut apart from their coffee mug."


An explicit construction of surfaces of genus g is given in the article on the fundamental polygon
Fundamental polygon

In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges....
. Image:Sphere-wireframe.png|genus 0 Image:Torus illustration.png|genus 1 Image:Double torus illustration.png|genus 2 Image:Triple torus illustration.png|genus 3

Non-orientable surface

The (non-orientable
Orientability

A surface S in the Euclidean space R3 is orientable if a two-dimensional figure cannot be moved around the surface and back to where it started so that it looks like its own mirror image ....
) genus of a connected, non-orientable closed surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 is a positive integer
Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
 representing the number of cross-cap
Cross-cap

In mathematics, a cross-cap is a two-dimensional surface that is homeomorphism to a M?bius strip. The term 'cross-cap', however, often implies that the surface has been deformed so that its boundary is an ordinary circle....
s attached to a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 ?, via the relationship ? = 2 - k, where k is the non-orientable genus.

For instance:
  • A projective plane
    Projective plane

    In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another coming from axiomatic geometry and finite geometry....
     has non-orientable genus one.
  • A Klein bottle
    Klein bottle

    In mathematics, the Klein bottle is a certain non-orientability surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the M?bius strip and the real projective plane....
     has non-orientable genus two.


Knot

The genus of a knot
Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations ....
 K is defined as the minimal genus of all Seifert surface
Seifert surface

In mathematics, a Seifert surface is a surface whose boundary of a manifold is a given knot or link . Such surfaces can be used to study the properties of the associated knot or link....
s 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
Handlebody

In the mathematics field of geometric topology, a handlebody is a decomposition of a manifold into standard pieces. Handlebodies play an important role in Morse theory, cobordism theory and the surgery theory of high-dimensional manifolds....
 is an integer representing the maximum number of cuttings along embedded disks
Disk (mathematics)

In geometry, a disk is the region in a plane bounded by a circle.A disk is said to be closed or open according to whether or not it contains the circle that constitutes its boundary....
 without rendering the resultant manifold disconnected. It is equal to the number of handles on it.

For instance:
  • A ball
    Ball (mathematics)

    In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general....
     has genus zero.
  • A solid torus has genus one.


Graph theory


The genus of a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 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
Planar graph

In graph theory, a planar graph is a graph which can be graph embedding in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints....
 has genus 0, because it can be drawn on a sphere without self-crossing.

The non-orientable genus of a graph
Graph (mathematics)

In mathematics a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges....
 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).

In topological graph theory
Topological graph theory

In mathematics topological graph theory is a branch of graph theory. It studies the embedding of graph s in surfaces, and graphs as topological spaces....
 there are several definitions of the genus of a group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
. Arthur T. White introduced the following concept. The genus of a group is the minimum genus of any of (connected, undirected) Cayley graph
Cayley graph

In mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph theory that encodes the structure of a discrete group....
s for .

The graph genus problem is NP-complete (Thomassen 1989).

Algebraic geometry


There are two related definitions of genus of any projective algebraic scheme
Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry....
 X: the arithmetic genus
Arithmetic genus

In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface....
 and the geometric genus
Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant p'g of algebraic varieties, defined for non-singular complex projective varieties as the Hodge number h'n,0 ....
. When X is a algebraic curve
Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension of an algebraic variety one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections....
 with field of definition the complex number
Complex number

In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies:...
s, and if X has no singular points
Tangent space

In mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
, then both of these definitions agree and coincide with the topological definition applied to the Riemann surface
Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold....
 of X (its manifold
Manifold

In mathematics, more specifically topology, a manifold is a topological space in which every point has a neighborhood which "resembles" Euclidean space....
 of complex points). The definition of elliptic curve
Elliptic curve

In mathematics, an elliptic curve is a differentiable manifold, algebraic variety#Projective varieties algebraic curve of genus #Algebraic geometry one, on which there is a specified point O....
 from algebraic geometry is non-singular curve of genus 1 with a given point on it.

See also

Cayley graph
Cayley graph

In mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph theory that encodes the structure of a discrete group....
 Group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 Geometric genus
Geometric genus

In algebraic geometry, the geometric genus is a basic birational invariant p'g of algebraic varieties, defined for non-singular complex projective varieties as the Hodge number h'n,0 ....
 Arithmetic genus
Arithmetic genus

In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface....
 Genus of a multiplicative sequence
Genus of a multiplicative sequence

In mathematics, the genus of a multiplicative sequence is a ring homomorphism, from the cobordism of smooth oriented compact manifolds to another ring , usually the ring of rational numbers....
 **