Kind (mathematics)

See also: Kind

Topology

Surface closed

The kind of a algebraic Curve, i.e. of a Surface (i.e a topological space of which any point has a homeomorphic vicinity in the plan) related, is the maximum number of simple curves closed without common points being able to be traced inside this surface without disconnecting it. In other words, in the process of determination of the kind, the complement of these curves remains related.

More concretely, if it is considered that surface is out of paper, the kind is the maximum number of feasible closed cuttings without surface not being separate in several pieces.

It is a concept of Topologie: two surfaces not having the same kind are not homeomorphic S.

Examples

  • a Sphere and a disc have a kind of 0.

  • a Tore has a kind of 1.

Not-singular curve

The kind of a not-singular curve in the projective Space \ mathbb P^2 (\ mathbb C) is defined by g= (n-1) (N2) /2 where the curve is defined by an irreducible polynomial of degree N, p (X, there) =0 .

Examples

  • a elliptic Courbe has a kind of 1.

Theory of the nodes

The kind of a node is half of the minimal number of handle S which it is necessary to add to the Sphère in order to be able to trace a line of cutting on its surface with the node, so that this one is divided into two at the time of cutting.

Examples

  • a Nœud of clover is of kind 1.

  • a commonplace Nœud is of kind 0.

Graph theory

The kind of a graph is the smallest entirety p so that the graph is Représentable on a directional Surface of p. kind.

Examples

  • the planar graphs are of kind 0.

  • the Graphe of the wells of the three houses is of kind 1.

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