the genus of a surface
/ JEE-nus /
The genus of a surface is, in the friendliest terms, the number of holes — or, said more carefully, the number of handles. A sphere has genus 0: no holes, no handles. A doughnut (torus) has genus 1: one hole you can poke a finger through, equivalently one handle. A two-holed doughnut, like a pretzel-ish double torus, has genus 2, and so on. The genus is the single counting number that, together with orientability, completely pins down which compact surface you are looking at.
To make 'number of holes' precise without waving your hands, topologists use one of two equivalent descriptions. One: the genus g is the number of handles you would attach to a sphere to build the surface — start with a sphere and glue on g handles (tubes), and you have the genus-g orientable surface. Two, more operationally: the genus is the largest number of disjoint closed curves you can draw on the surface and cut along without splitting it into two pieces. On a sphere any closed curve cuts it in two (genus 0); on a torus you can cut along one loop — say around the tube — and still have one connected piece, but a second independent such cut would separate it (genus 1).
Genus is a topological invariant: it cannot change under any stretching or bending, so it certifies that, say, a sphere and a torus are genuinely different surfaces — no homeomorphism can turn 0 holes into 1. It connects directly to the Euler characteristic by chi = 2 - 2g for a closed orientable surface, which is how you can compute the genus by counting vertices, edges, and faces of any triangulation. The honest caveat: 'number of holes' is intuitive but slippery in casual speech — the genus counts handles (holes you can thread through), and a hollow sphere, despite enclosing empty space, has genus 0 because that enclosed cavity is not a handle.
A coffee mug has genus 1: ignore the bowl part (which is just a dent, deformable away) and what remains topologically is the handle, a single tube — exactly one hole to thread, like a doughnut. A pair of eyeglass frames (two lens rings joined by a bridge) is genus 2: two holes. And a button with four thread-holes is genus 3 once you account for the disk it sits in — every drilled-through hole adds one to the genus.
Genus counts threadable holes: mug 1, eyeglass frames 2, four-holed button 3.
Genus counts handles (holes you can thread something through), not enclosed cavities: a hollow ball encloses empty space yet has genus 0, the same as a solid sphere's surface.