 Kirby calculus

For integral Dehn surgery on a framed link in $S^3$, Kirby calculus uses two moves: a handle slide and addition or removal of a split unknot with framing $+1$ or $-1$. Two surgery diagrams give orientation-preservingly diffeomorphic closed three-manifolds exactly when these moves and link isotopy relate them.^[ref]

A handle slide replaces one attaching component by its band sum with a framed push-off of another; on the surgery linking matrix, it performs the matching integral row and column operation. It preserves the four-dimensional surgery trace. Adding a split $\pm 1$-framed unknot preserves the boundary three-manifold but changes the trace by connected sum with ${\mathbb{CP}}^2$ or its orientation reverse. This differs from creation or cancellation of a canceling handle pair in a four-dimensional handle decomposition, which preserves the four-dimensional manifold itself.

Kirby calculus is a mathematical technique used in the field of low-dimensional topology, particularly in the study of 3-manifolds. It is named after Rob Kirby, who introduced this concept in a series of papers in the 1970s. The main focus of Kirby calculus is on the manipulation and understanding of 3-manifolds via the use of specific types of diagrams called Kirby diagrams or handlebody diagrams.