Dehn Surgery

The operation of drilling a tubular Neighborhood of a Knot $K$ in $\Bbb{S}^3$ and then gluing in a solid Torus so that its meridian curve goes to a $(p,q)$-curve on the Torus boundary of the Knot exterior. Every compact connected 3-Manifold comes from Dehn surgery on a Link in $\Bbb{S}^3$.

See also Kirby Calculus

References

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, p. 260, 1994.