Kirby calculus

Summary

Kirby calculus is a set of moves on framed link diagrams in the $3$-sphere that relate any two diagrams describing the same $3$-manifold. A framed link describes a $4$-manifold, obtained by attaching $2$-handles to the $4$-ball along the link, and its boundary $3$-manifold, obtained by Dehn surgery. Robion Kirby proved that two framed links give the same oriented $3$-manifold if and only if they are related by his moves. It is a basic working tool of $3$- and $4$-manifold topology, used to construct manifolds and to prove that two of them are diffeomorphic.