Genus
Geometry and topology
For a closed orientable surface, the number of holes: a sphere has genus 0, a torus genus 1, a two-holed torus genus 2.
What is usually left out
Genus classifies closed orientable surfaces completely — two are homeomorphic exactly when their genus agrees, which is unusually clean for a topological invariant. It also connects to the Euler characteristic by χ = 2 − 2g, and it is the invariant that makes an elliptic curve "genus 1".