In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. [1] A sphere has genus 0, while a torus has genus 1.
In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. [1] A sphere has genus 0, while a torus has genus 1.
The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. [2] 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.
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). [3] 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:
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. [4] 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:
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. [5]
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. [6]
There are two related definitions of genus of any projective algebraic scheme X: the arithmetic genus and the geometric genus. [7] When X is an algebraic curve with field of definition the complex numbers, and if X has no singular points, then these definitions agree and coincide with the topological definition applied to the Riemann surface of X (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, an irreducible plane curve of degree given by the vanishing locus of a section has geometric genus
where is the number of singularities when properly counted.
In differential geometry, a genus of an oriented manifold may be defined as a complex number subject to the conditions
In other words, is a ring homomorphism , where is Thom's oriented cobordism ring. [8]
The genus is multiplicative for all bundles on spinor manifolds with a connected compact structure if is an elliptic integral such as for some This genus is called an elliptic genus.
The Euler characteristic is not a genus in this sense since it is not invariant concerning cobordisms.
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. [9]
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solid figures; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.
In geometry, a torus is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut.
In the mathematical field of differential geometry, the Gauss–Bonnet theorem is a fundamental formula which links the curvature of a surface to its underlying topology.
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry.
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by .
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 versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index. It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
In the mathematical field of geometric topology, a Heegaard splitting is a decomposition of a compact oriented 3-manifold that results from dividing it into two handlebodies.
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime. A torus knot is trivial if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a -bundle over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure.
In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants.
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf.
In topological graph theory, an embedding of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs are associated with edges in such a way that:
In mathematics, a regular map is a symmetric tessellation of a closed surface. More precisely, a regular map is a decomposition of a two-dimensional manifold into topological disks such that every flag can be transformed into any other flag by a symmetry of the decomposition. Regular maps are, in a sense, topological generalizations of Platonic solids. The theory of maps and their classification is related to the theory of Riemann surfaces, hyperbolic geometry, and Galois theory. Regular maps are classified according to either: the genus and orientability of the supporting surface, the underlying graph, or the automorphism group.
In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces.
In mathematics, a genus g surface is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g distinct tori and the boundaries of the g many disks are identified, forming a g-torus. The genus of such a surface is g.
