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 chi. 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).

See also

Curve Genus, Euler Characteristic, Graph Genus, Knot Genus

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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.

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Cite this as:

Weisstein, Eric W. "Genus." From MathWorld--A Wolfram Web Resource.

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