In topology, a means of combining surfaces or, more generally, manifolds of the same dimension. Given two closed surfaces S₁ and S₂, each with a triangulation, their connected sum S₁#S₂ is formed by removing a triangle from each and gluing the surfaces along the missing triangles’ boundaries (as an identification space). The Euler characteristic of S₁#S₂ satisfies

For example, the connected sum of two tori is a torus with two holes, as in the figure.

Connected sum of two tori

If the two manifolds are oriented, then their connected sum is unique up to homeomorphism. In a similar manner, the knot sum of two knots can be made. If both knots are oriented, then their knot sum is well defined up to isotopy. See Classification Theorem for Surfaces, prime knot.