Genus (mathematics) - Definition and Overview

In mathematics, the genus has few different meanings

Topology

The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it.

For instance:

• A sphere, disc and annulus all have genus zero.
• A torus has genus one, as does the surface of a coffee cup.

The genus of a knot <math>K<math> is the least integer <math>g=g(K)<math> such that <math>K<math> is the boundary of a Seifert surface of genus <math>g<math>.

For instance:

• An unknot (also called a trivial knot) <math>O<math>–which is, by definition, the boundary of a disc embedded in the <math>3<math>-sphere <math>S^3<math>–has genus zero, and any knot of genus zero is an unknot.
• The trefoil knot has genus one, as does the figure-eight knot.

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:

• A ball has genus zero.
• A solid torus <math>D^2\times S^1<math> has genus one.

Graph theory

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 genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. How about this in matrix theory? could anyone say....

Algebraic geometry

There is a definition of genus of any algebraic curve C. When the field of definition for C is the complex numbers, and C has no singular points, then that definition coincides with the topological definition applied to the Riemann surface of C (its manifold of complex points). The definition of elliptic curve from algebraic geometry is non-singular curve of genus 1.