# Genus (mathematics)

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In mathematics, genus (plural genera) has a few different, but closely related, meanings:

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### Orientable surface

The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple 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, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads χ = 2 − 2g − b.

For instance:

• The sphere S2 and a disc both have genus zero.
• A torus has genus one, as does the surface of a coffee mug with a handle. This is the source of the joke that "a topologist is someone who can't tell their donut apart from their coffee mug."

An explicit construction of surfaces of genus g is given in the article on the fundamental polygon.

genus 0

genus 1

genus 2

genus 3

In simpler terms, the value of an orientable surface's genus is equal to the number of "holes" it has.[1]

### Non-orientable surfaces

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere. Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.

For instance:

### Knot

The genus of a knot K is defined as the minimal genus of all Seifert surfaces for K. A Seifert surface of a knot is however a manifold with boundary the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.