Inmathematics,genus (pl.:genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of asurface.[1] Asphere has genus 0, while atorus has genus 1.
Thegenus of aconnected, orientable surface is aninteger representing the maximum number of cuttings along non-intersectingclosed simple curves without rendering the resultantmanifold disconnected.[2] It is equal to the number ofhandles on it. Alternatively, it can be defined in terms of theEuler characteristic, via the relationship forclosed surfaces, where is the genus. For surfaces withboundary components, the equation reads.
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] Atorus has 1 such hole, while asphere has 0. The green surface pictured above has 2 holes of the relevant sort.
For instance:
Explicit construction ofsurfaces of the genusg is given in the article on thefundamental polygon.
Thenon-orientable genus,demigenus, orEuler genus of a connected, non-orientable closed surface is a positive integer representing the number ofcross-caps attached to asphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 −k, wherek is the non-orientable genus.
For instance:
Thegenus of aknotK is defined as the minimal genus of allSeifert surfaces forK.[4] A Seifert surface of a knot is however amanifold with boundary, the boundary being the knot, i.e.homeomorphic to theunit 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.
Thegenus of a 3-dimensionalhandlebody is an integer representing the maximum number of cuttings along embeddeddisks without rendering the resultant manifold disconnected. It is equal to the number of handles on it.
For instance:
Thegenus of agraph is the minimal integern such that the graph can be drawn without crossing itself on a sphere withn handles (i.e. an oriented surface of the genusn). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing.
Thenon-orientable genus of agraph is the minimal integern such that the graph can be drawn without crossing itself on a sphere withn cross-caps (i.e. a non-orientable surface of (non-orientable) genusn). (This number is also called thedemigenus.)
TheEuler genus is the minimal integern such that the graph can be drawn without crossing itself on a sphere withn cross-caps or on a sphere withn/2 handles.[5]
Intopological graph theory there are several definitions of the genus of agroup. Arthur T. White introduced the following concept. The genus of a groupG is the minimum genus of a (connected, undirected)Cayley graph forG.
Thegraph genus problem isNP-complete.[6]
There are two related definitions ofgenus of anyprojective algebraicscheme: thearithmetic genus and thegeometric genus.[7] When is analgebraic curve withfield of definition thecomplex numbers, and if has nosingular points, then these definitions agree and coincide with the topological definition applied to theRiemann surface of (itsmanifold of complex points). For example, the definition ofelliptic curve fromalgebraic geometry isconnected non-singular projective curve of genus 1 with a givenrational point on it.
By theRiemann–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.
Indifferential geometry, a genus of anoriented manifold may be defined as a complex number subject to the conditions
In other words, is aring homomorphism, where is Thom'soriented cobordism ring.[8]
The genus is multiplicative for all bundles onspinor manifolds with a connected compact structure if is anelliptic 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 innucleic acids orproteins. 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]
