The genus of a closed connected surface is its number of handles when it is an orientable surface and its number of cross-caps when it is a nonorientable surface. It is a topological invariant that roughly measures the number of holes in the surface.
The genus of a surface, also called the geometric genus, is related to the Euler characteristic 
. For a orientable
surface such as a sphere (genus 0) or torus
(genus 1), the relationship is
 |
For a nonorientable surface such as a real projective plane (genus 1) or Klein bottle (genus 2), the relationship is
 |
(Massey 1997). The corresponding graph invariants are the orientable genus, nonorientable genus, and Euler genus.
