= Kirby calculus
{c}
{wiki}
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.<A calculus for framed links in S3 by Robion Kirby>[Section 1, Operations One and Two and Theorem 1, pages 35–36]{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 $\pm1$-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.
Back to article page
