In mathematics, genus (: 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.
\chi
\chi=2-2g
g
b
\chi=2-2g-b
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).[1] 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:
S2
Explicit construction of surfaces of the genus g is given in the article on the fundamental polygon.
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:
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.
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:
See main article: Graph embedding.
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 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.[2]
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.[3]
X
X
X
X
By the Riemann–Roch theorem, an irreducible plane curve of degree
d
s\in\Gamma(P2,
| l{O} | |
| P2 |
(d))
| g= | (d-1)(d-2) |
| 2 |
-s,
where
s
M
\Phi(M)
\Phi(M1\amalgM2)=\Phi(M1)+\Phi(M2)
\Phi(M1 x M2)=\Phi(M1) ⋅ \Phi(M2)
\Phi(M1)=\Phi(M2)
M1
M2
In other words,
\Phi
R\toC
R
The genus
\Phi
log\Phi
log\Phi
| x | |
| (x)=\int | |
| 0 |
(1-2\deltat2+\varepsilont4)-1/2dt
\delta,\varepsilon\inC.
The Euler characteristic
\chi(M)
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.[6]