Жүзінің "тесіктерінің" саны

Number of "holes" of a surface

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.

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 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. 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.

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.