---
tag: NJ787GZS1N
kind: concept
name: Genus
msc: [57K20]
sources:
  - url: https://en.wikipedia.org/wiki/Genus_(mathematics)
---

## Summary

The genus of a closed orientable [[surface|VGJJVBZ1MQ]] is its number of handles.
It is the largest number of disjoint simple closed curves along which the surface can be cut without falling apart.
By the [[classification of surfaces|2SFM73YET0]], the genus determines a closed orientable surface up to homeomorphism, and it is related to the [[Euler characteristic|RNDDWDAFJ5]] by $\chi = 2 - 2g$.
The same number appears as the genus of an algebraic curve and of a compact Riemann surface.

## Explanations

### L1 human

The genus counts holes in the sense of a doughnut, not of a cup.
A sphere has genus $0$, a doughnut has genus $1$, and a pretzel with two holes has genus $2$.
On a doughnut you can cut along one circle around the tube and still have one piece, a cylinder.
Any second disjoint cut then splits it, so the genus is $1$.