Connected Sum

The connected sum is an operation in topology that combines two manifolds of the same dimension into a single new manifold by removing a small ball from each and gluing the resulting boundary spheres together. For surfaces, repeated connected sums of tori or projective planes with a sphere generate every closed orientable or non-orientable surface, which is the basis of the classification theorem for closed surfaces. The operation generalizes to higher-dimensional manifolds and to knots, where the connected sum of two knots forms a new, generally more complicated knot.