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).

See also Curve Genus ,

Euler Characteristic ,

Graph Genus ,

Knot
Genus
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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. Referenced on Wolfram|Alpha Genus
Cite this as:
Weisstein, Eric W. "Genus." From MathWorld --A
Wolfram Web Resource. https://mathworld.wolfram.com/Genus.html

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