In mathematics, genus has a few different, but closely related, meanings:
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Topology
Orientable surface
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object It is equal to the number of handles on it. In Topology, a branch of Mathematics, a handle is just a topological ball; it is called a handle because of the context in which it is discussed of which Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant For surfaces with b boundary components, the equation reads χ = 2 − 2g − b.
For instance:
- A sphere, disc and annulus all have genus zero. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. In Mathematics, an annulus (the Latin word for "little ring" with plural annuli) is a ring-shaped geometric figure or more generally a term
- A torus has genus one, as does the surface of a coffee mug with a handle. In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar 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. In Mathematics, each closed Surface in the sense of Geometric topology can be constructed from an even-sided oriented Polygon, called a fundamental
 genus 0 |
 genus 1 |
 genus 2 |
 genus 3 |
Non-orientable surface
The (non-orientable) genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a cross-cap is a two-dimensional surface that is topologically equivalent to a Möbius strip. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe 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. In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant
For instance:
- A projective plane has non-orientable genus one. See Real projective plane and Complex projective plane, for the cases met as manifolds of respective dimension 2 and 4 In Mathematics
- A Klein bottle has non-orientable genus two. In Mathematics, the Klein bottle is a certain non- orientable Surface, i
Knot
The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. In Mathematics, a Seifert surface is a Surface whose boundary is a given knot or link. In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations In Mathematics, a Seifert surface is a Surface whose boundary is a given knot or link.
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. In the mathematical field of Geometric topology, a handlebody is a particular kind of Manifold. In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. It is equal to the number of handles on it.
For instance:
- A ball has genus zero. 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
- A solid torus
has genus one.
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. In Mathematics and Computer science, graph theory is the study of graphs: mathematical structures used to model pairwise relations between objects In Topological graph theory, an embedding of a graph G on a Surface &Sigma is a representation of G on &Sigma in which points In Mathematics and Computer science, a graph is the basic object of study in Graph theory. e. an oriented surface of genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. Kuratowski's and Wagner's theorems The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of Forbidden
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. In Mathematics and Computer science, a graph is the basic object of study in Graph theory. e. a non-orientable surface of (non-orientable) genus n).
In topological graph theory there are several definitions of the genus of a group. In Mathematics topological graph theory is a branch of Graph theory. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Arthur T. White introduced the following concept. The genus of a group G is the minimum genus of any of (connected, undirected) Cayley graphs for G. In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group.
Algebraic geometry
There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, the arithmetic genus of an Algebraic variety is one of some possible generalizations of the Genus of an algebraic curve or Riemann In Algebraic geometry, the geometric genus is a basic Birational invariant p g of Algebraic varieties, defined for When X is a algebraic curve with field of definition the complex numbers, and if X has no singular points, then both of these definitions agree and coincide with the topological definition applied to the Riemann surface of X (its manifold of complex points). In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted 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 Mathematics, particularly in Complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The definition of elliptic curve from algebraic geometry is non-singular curve of genus 1 with a given point on it. In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O
See also
In Mathematics, the Cayley graph, also known as the Cayley colour graph, is the graph that encodes the structure of a Discrete group. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebraic geometry, the geometric genus is a basic Birational invariant p g of Algebraic varieties, defined for In Mathematics, the arithmetic genus of an Algebraic variety is one of some possible generalizations of the Genus of an algebraic curve or Riemann In Mathematics, the genus of a multiplicative sequence is a Ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to anotherDapyx Software network: MP3 Explorer | Ebook Manager | Zenithic