Editing Genus (mathematics)

{{Short description|Number of "holes" of a surface}}[[File:Double torus illustration.png|thumb|A genus-2 surface]]

In [[mathematics]], '''genus''' ({{plural form}}: '''genera''') has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a [[Surface (topology)|surface]].{{sfn|Popescu-Pampu|2016|loc=Introduction|p=xiii}} A [[sphere]] has genus 0, while a [[torus]] has genus 1.

==Topology==

===Orientable surfaces===<!-- This section is linked from [[Complex plane]] -->
[[File:Mug and Torus morph.gif|thumb|The coffee cup and donut shown in this animation both have genus one.]]
The '''genus''' of a [[Connected space|connected]], orientable surface is an [[integer]] representing the maximum number of cuttings along non-intersecting [[Curve#Topological_curve|closed simple curves]] without rendering the resultant [[manifold]] disconnected.{{sfn|Popescu-Pampu|2016|loc=Introduction|p=xiv}} It is equal to the number of [[Handle (mathematics)|handles]] on it. Alternatively, it can be defined in terms of the [[Euler characteristic]] <math>\chi</math>, via the relationship <math>\chi=2-2g</math> for [[Surface_(topology)#Closed_surfaces|closed surfaces]], where <math>g</math> is the genus. For surfaces with <math>b</math> [[Boundary (topology)|boundary]] components, the equation reads <math>\chi=2-2g-b</math>.

In layman's terms, the genus is 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).<ref>{{Cite web | url=http://mathworld.wolfram.com/Genus.html | title=Genus | last = Weisstein | first = E.W. | website = MathWorld | access-date = 4 June 2021 }}</ref> 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]] <math>S^2</math> and a [[disk (mathematics)|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 [[Genus g surface|surfaces of the genus ''g'']] is given in the article on the [[fundamental polygon]].

<gallery widths="100" heights="100" perrow="4" mode="nolines" caption="Genus of orientable surfaces">
File:Green Sphere illustration.png|[[Planar graph]]: genus 0 
File:Torus illustration.png|[[Toroidal graph]]: genus 1
File:Double torus illustration.png|[[Teapot]]: Double Toroidal graph: genus 2
File:Triple torus illustration.png|[[Pretzel]] graph: genus 3
</gallery>

===Non-orientable surfaces===
The '''[[orientability|non-orientable]] genus''', '''demigenus''', or '''Euler genus''' of a connected, non-orientable closed surface is a positive integer representing the number of [[cross-cap]]s 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]] has non-orientable genus 2.

===Knot===
The '''[[genus of a knot|genus]]''' of a [[knot (mathematics)|knot]] ''K'' is defined as the minimal genus of all [[Seifert surface]]s for ''K''.<ref>{{Citation|first=Colin |last= Adams|author-link=Colin Adams (mathematician)|title=The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots |publisher=[[American Mathematical Society]]|year=2004|isbn=978-0-8218-3678-1}}</ref> A Seifert surface of a knot is however a [[manifold with boundary]], the boundary being the knot, i.e.
[[Homeomorphism|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 [[Disk (mathematics)|disks]] without rendering the resultant manifold disconnected. It is equal to the number of handles on it.

For instance:
* A [[Ball (mathematics)|ball]] has genus 0.
* A solid torus ''D''<sup>2</sup> × ''S''<sup>1</sup> has genus 1.

===Graph theory===
{{Main|Graph embedding}}

The '''genus''' of a [[Graph (discrete mathematics)|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 has genus 0, because it can be drawn on a sphere without self-crossing.

The '''non-orientable genus''' of a [[Graph (discrete mathematics)|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.<ref>{{cite book | last=Ellis-Monaghan | first=Joanna A. | last2=Moffatt | first2=Iain | title=Graphs on Surfaces: Dualities, Polynomials, and Knots | publisher=Springer New York | publication-place=New York, NY | date=2013 | isbn=978-1-4614-6970-4 | doi=10.1007/978-1-4614-6971-1 | page=}}</ref>

In [[topological graph theory]] there are several definitions of the genus of a [[Group (mathematics)|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 embedding#Computational complexity|graph genus problem]] is [[NP-complete]].<ref>{{cite journal|first1=Carsten |last1=Thomassen |title= The graph genus problem is NP-complete |journal= Journal of Algorithms |year=1989 |issue=4 |volume=10 |pages=568&ndash;576 |issn=0196-6774 |doi=10.1016/0196-6774(89)90006-0 | zbl=0689.68071 }}</ref>

==Algebraic geometry==
There are two related definitions of '''genus''' of any [[projective scheme|projective]] algebraic [[Scheme (mathematics)|scheme]] <math>X</math>: the [[arithmetic genus]] and the [[geometric genus]].<ref>{{cite book | last=Hirzebruch | first=Friedrich | author-link=Friedrich Hirzebruch | title=Topological methods in algebraic geometry | others=Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel | edition=Reprint of the 2nd, corr. print. of the 3rd | orig-year=1978 | series=Classics in Mathematics | location=Berlin | publisher=[[Springer-Verlag]] | year=1995 | isbn=978-3-540-58663-0 | zbl=0843.14009 }}</ref>  When <math>X</math> is an [[algebraic curve]] with [[Field (mathematics)|field]] of definition the [[complex number]]s, and if <math>X</math> has no [[tangent space|singular points]], then these definitions agree and coincide with the topological definition applied to the [[Riemann surface]] of <math>X</math> (its [[manifold]] of complex points). For example, the definition of [[elliptic curve]] from [[algebraic geometry]] is ''connected non-singular projective curve of genus 1 with a given [[rational point]] on it''.

By the [[Riemann–Roch theorem#Applications|Riemann–Roch theorem]], an irreducible plane curve of degree <math>d</math> given by the vanishing locus of a section <math>s \in \Gamma(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(d))</math> has geometric genus

:<math>g=\frac{(d-1)(d-2)}{2}-s,</math>

where <math>s</math> is the number of singularities when properly counted.

==Differential geometry==

In [[differential geometry]], a genus of an [[oriented manifold]] <math>M</math> may be defined as a complex number <math>\Phi(M)</math> subject to the conditions
* <math>\Phi(M_{1}\amalg M_{2})=\Phi(M_{1})+\Phi(M_{2})</math>
* <math>\Phi(M_{1}\times M_{2})=\Phi(M_{1})\cdot \Phi(M_{2})</math>
* <math>\Phi(M_{1})=\Phi(M_{2})</math> if <math>M_{1}</math> and <math>M_{2}</math> are [[cobordant]]. 

In other words, <math>\Phi</math> is a [[ring homomorphism]] <math>R\to\mathbb{C}</math>, where <math>R</math> is Thom's [[oriented cobordism ring]].<ref>Charles Rezk - Elliptic cohomology and elliptic curves (Felix Klein lectures, Bonn 2015. Department of Mathematics, University of Illinois, Urbana, IL)</ref>

The genus <math>\Phi</math> is multiplicative for all bundles on [[spinor]] manifolds with a connected compact structure if <math>\log_{\Phi}</math> is an [[elliptic integral]] such as <math>\log_{\Phi}(x)=\int^{x}_{0}(1-2\delta t^{2}+\varepsilon t^{4})^{-1/2}dt</math> for some <math>\delta,\varepsilon\in\mathbb{C}.</math> This genus is called an elliptic genus. 

The Euler characteristic <math>\chi(M)</math> 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 acid]]s or [[protein]]s. 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.<ref>{{Cite journal|last1=Sułkowski|first1=Piotr|last2=Sulkowska|first2=Joanna I.|last3=Dabrowski-Tumanski|first3=Pawel|last4=Andersen|first4=Ebbe Sloth|last5=Geary|first5=Cody|last6=Zając|first6=Sebastian|date=2018-12-03|title=Genus trace reveals the topological complexity and domain structure of biomolecules|journal=Scientific Reports|language=en|volume=8|issue=1|pages=17537|doi=10.1038/s41598-018-35557-3|issn=2045-2322|pmc=6277428|pmid=30510290|bibcode=2018NatSR...817537Z}}</ref>

==See also==
* [[Group (mathematics)]]
* [[Arithmetic genus]]
* [[Geometric genus]]
* [[Genus of a multiplicative sequence]]
* [[Genus of a quadratic form]]
* [[Spinor genus]]

==Citations==
{{Reflist}}

==References==
{{refbegin}}
*{{cite book|first=Patrick|last=Popescu-Pampu|title=What is the Genus?|year=2016|publisher=[[Springer Verlag]]|url=https://www.springer.com/gp/book/9783319423111|isbn=978-3-319-42312-8}}
{{refend}}

{{DEFAULTSORT:Genus (Mathematics)}}
[[Category:Topology]]
[[Category:Geometric topology]]
[[Category:Surfaces]]
[[Category:Algebraic topology]]
[[Category:Algebraic curves]]
[[Category:Graph invariants]]
[[Category:Topological graph theory]]
[[Category:Geometry processing]]

{{Set index article}}