Dehn surgery

Summary

Dehn surgery builds a new 3-manifold from a knot or link in a given one. One removes a solid torus around the knot. Then one glues it back so that its meridian disc caps off a chosen curve on the boundary torus. For knots in $S^3$ the choices are labelled by a number $p/q \in \Q \cup \set{\infty}$. By the Lickorish–Wallace theorem, every closed orientable $3$-manifold arises by Dehn surgery on a link in $S^3$.