Genus Of A Curve
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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 S2 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 d given by the vanishing locus of a section s \in \Gamma(\mathbb^2, \mathcal_(d)) has geometric genus :g=\frac-s, where ''s'' is the number of singularities when properly counted.


Differential geometry

In differential geometry, a genus of an oriented manifold M may be defined as a complex number \Phi(M) subject to the conditions * \Phi(M_\amalg M_)=\Phi(M_)+\Phi(M_) * \Phi(M_\times M_)=\Phi(M_)\cdot \Phi(M_) * \Phi(M_)=\Phi(M_) if M_ and M_ 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, \Phi is a ring homomorphism R\to\mathbb, where R is Thom's oriented cobordism ring. The genus \Phi is multiplicative for all bundles on spinor manifolds with a connected compact structure if \log_ is an elliptic integral such as \log_(x)=\int^_(1-2\delta t^+\varepsilon t^)^dt for some \delta,\varepsilon\in\mathbb. This genus is called an elliptic genus. The Euler characteristic \chi(M) 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 {{Set index article