Genus g surface

In mathematics, a genus g surface (also known as a g-torus or g-holed torus) is a surface formed by the connected sum of g distinct tori: the interior of a disk is removed from each of g distinct tori and the boundaries of the g many disks are identified (glued together), forming a g-torus. The genus of such a surface is g.

A genus g surface is a

The genus of a connected orientable surface is an

An orientable surface of genus zero is the sphere S². Another surface of genus zero is the disc.

• Representations of genus 0 surfaces
• 
  A sphere ${\displaystyle S^{2}}$
• 
  A closed disc (with boundary)

Genus 1

A genus one orientable surface is the ordinary torus. A non-orientable surface of genus one is the projective plane.^[2]

• Representations of genus 1 surfaces
• 
  A torus of genus 1
• 
  An elliptic curve

Genus 2

The term double torus is occasionally used to denote a genus 2 surface.^[4] A non-orientable surface of genus two is the Klein bottle.

The Bolza surface is the most symmetric Riemann surface of genus 2, in the sense that it has the largest possible conformal automorphism group.^[5]

• Representations of genus 2 surfaces
• 
  A torus of genus 2

Genus 3

The term triple torus is also occasionally used to denote a genus 3 surface.^[6]

The Klein quartic is a compact Riemann surface of genus 3 with the highest possible order automorphism group for compact Riemann surfaces of genus 3. It has 168 orientation-preserving automorphisms, and 336 automorphisms altogether.

• Several genus 3 surfaces
• 
  A sphere with three

  handles
• 
  The

  tori
• 
  Triple torus
• 
  Dodecagon with opposite edges identified^[7]
• 
  Tetradecagon with opposite edges identified^[7]

See also

• Three-torus
• g-torus knot

References