genus (mathematics)
{{dablink|For "genus" in index theory, see
Genus of a multiplicative sequence.}}In
mathematics,
genus has a few different, but closely related, meanings:
Topology
Euler characteristic χ, via the relationship
χ 2 − 2g for closed surfaces, where
g is the genus. For surfaces with
b boundary components, the equation reads
χ 2 − 2g − b.">
Orientable surfaceThe 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. It is equal to the number of handles on it. 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. For surfaces with b boundary components, the equation reads χ 2 − 2g − b.
For instance:
- A sphere, disc and annulus all 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 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.
Image:Sphere-wireframe.png|genus 0Image:Torus illustration.png|genus 1Image:Double torus illustration.png|genus 2Image:Triple torus illustration.png|genus 3Non-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. 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:
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 has genus zero.
- A solid torus
D2⋅ S1
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.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.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).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 any of (connected, undirected)
Cayley graphs for
G
.The graph genus problem is NP-complete (Thomassen 1989).
Algebraic geometry
There are two related definitions of
genus of any projective algebraic
scheme X: the
arithmetic genus and the
geometric genus. 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). The definition of
elliptic curve from algebraic geometry is
non-singular curve of genus 1 with a given point on it.
See also
GenusGeschlecht (Fläche)Genro (matematiko)Genre (mathématiques)Genere (matematica)Genus (wiskunde)GenusGénero (matemática)Matematiskt genus亏格
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