Genus (mathematics)

In mathematics, genus (pl.: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1.

Topology

Orientable surfaces

The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting simple closed curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic

χ

{\displaystyle \chi }

, via the relationship

χ

=

2

−

2

g

{\displaystyle \chi =2-2g}

for closed surfaces, where

g

{\displaystyle g}

is the genus. For surfaces with

b

{\displaystyle b}

boundary components, the equation reads

χ

=

2

−

2

g

−

b

{\displaystyle \chi =2-2g-b}

.

In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the relevant sort.