In mathematics, the genus of a topological space such as a surface is an integer representing the maximum number of cuts that can be made through it without rendering it disconnected. This is roughly equivalent to the number of holes in it, or handles on it.

For instance:

  • A point, line, and a sphere all have genus zero
  • A torus has genus one, as does a coffee cup as a solid object (solid torus), a Möbius strip, and the symbol 0.
  • The symbols 8 and B have genus two.
  • A pretzel has genus three.

In algebraic geometry there is a definition for the 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.

in graph theory, the genus of a graph is a integer n such that the graph can be drawn without crossing itself on a surface with n-handles. Thus, a planar graph has genus 0. (can be drawn on a sphere without self-crossing)