Editing Genus (mathematics) (section)

===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>