In mathematics, a genus ''g'' surface (also known as a ''g''-torus or ''g''-holed torus) is a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
formed by the
connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of ''g'' many
tori: the interior of a disk is removed from each of ''g'' many tori and the boundaries of the ''g'' many disks are identified (glued together), forming a ''g''-torus. The
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
of such a surface is ''g''.
A genus ''g'' surface is a
two-dimensional
In mathematics, a plane is a Euclidean (flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as s ...
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 ...
. The
classification theorem for surfaces states that every
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
two-dimensional manifold is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to either the sphere, the connected sum of tori, or the connected sum of
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
s.
Definition of genus
The genus of a connected orientable surface is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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
A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following tra ...
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.
The genus (sometimes called the 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 − ''g'', where ''g'' is the non-orientable genus.
Genus 0
An
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
surface of genus zero is the
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
''S''
''2''. Another surface of genus zero is the
disc.
File:Sphere filled blue.svg, A sphere
File:1-ball.svg, A closed disc (with boundary)
Genus 1
A genus one orientable surface is the ordinary torus. A non-orientable surface of genus one is the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ...
.
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s over the complex numbers can be identified with genus 1 surfaces. The formulation of elliptic curves as the embedding of a
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
in the
complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates
:(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1, ...
follows naturally from a property of
Weierstrass's elliptic functions
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by th ...
that allows elliptic curves to be obtained from the quotient of the
complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
by a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
.
File:Torus illustration.png, A torus of genus 1
File:Elliptic curve simple.png, An elliptic curve
Genus 2
The term double torus is occasionally used to denote a genus 2 surface.
A non-orientable surface of genus two is the
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 ...
.
The
Bolza surface
In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL_2(3) of order 48 ...
is the most symmetric
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
2, in the sense that it has the largest possible
conformal automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
.
File:Double torus illustration.png, A torus of genus 2
Genus 3
The term triple torus is also occasionally used to denote a genus 3 surface.
The
Klein quartic
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
is a compact
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
with the highest possible order
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
for compact Riemann surfaces of genus 3. It has namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed.
File:Sphere with three handles.png, A sphere with three handles
A handle is a part of, or attachment to, an object that allows it to be grasped and manipulated by hand. The design of each type of handle involves substantial ergonomic issues, even where these are dealt with intuitively or by following tra ...
File:Triple torus array.png, The connected sum
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...
of three tori
File:Triple torus illustration.png, Triple torus
File:Dodecagon with opposite faces identified.svg, Dodecagon
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
Regular dodecagon
A regular dodecagon is a figure with sides of the same length and internal angles of the same size. It has twelve lines of reflective symmetry and rotational sym ...
with opposite edges identified[Jürgen Jost, (1997) "Compact Riemann Surfaces: An Introduction to Contemporary Mathematics", Springer]
File:14-gon with opposite faces identified.svg, Tetradecagon
In geometry, a tetradecagon or tetrakaidecagon or 14-gon is a fourteen-sided polygon.
Regular tetradecagon
A '' regular tetradecagon'' has Schläfli symbol and can be constructed as a quasiregular truncated heptagon, t, which alternates two t ...
with opposite edges identified
See also
*
Three-torus
The three-dimensional torus, or 3-torus, is defined as any topological space that is homeomorphic to the Cartesian product of three circles, \mathbb^3 = S^1 \times S^1 \times S^1. In contrast, the usual torus is the Cartesian product of only two ...
*
g-torus knot
References
Sources
* James R. Munkres, ''Topology, Second Edition'', Prentice-Hall, 2000, {{ISBN, 0-13-181629-2.
* William S. Massey, ''Algebraic Topology: An Introduction'', Harbrace, 1967.
Topology
Geometry
Surfaces