Genus

A topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a 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 2003).

REFERENCES:

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, p. 635, 1997.

Massey, W. S. A Basic Course in Algebraic Topology. New York: Springer-Verlag, p. 30, 1997.