Genus

The genus of a immersed surface in ${\mathbb{R}}^3$ is, according to Riemann, the maximum number of cuts along closed, nonintersecting loops on the surface that can be performed without splitting the surface into two disconnected pieces. Needham_2021 page 166.

It can be computed according to the global Gauss-Bonnet theorem.

It classifies, topologically, the compact closed surfaces.