In mathematics, a genus-2 surface (also known as a double torus or two-holed torus) is a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 formed by the connected sum

Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...

 of two tori

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

. That is to say, from each of two tori the interior of a disk is removed, and the boundaries of the two disks are identified (glued together), forming a double torus.

This is the simplest case of the connected sum of n tori. A connected sum of tori is an example of a two dimensional manifold. According to the classification theorem

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 for 2-manifolds, every compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 connected

Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

 2-manifold is either a sphere, a connected sum of tori, or a connected sum of real projective plane

Real projective plane

In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...

s.

Double torus knot

Double torus knot

A double torus knot is a closed curve drawn on the surface called a double torus . More technically, a double torus knot is the homeomorphic image of a circle in S³ which can be realized as a subset of a genus two handlebody in S³...

s are studied in knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

.

Example

The Bolza surface

Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve , is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely 48. An affine model for the Bolza surface can be obtained as the locus of the...

 is the most symmetric hyperbolic surface of genus

Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:-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...

2.