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.

For an m-component integral framed surgery link, normalize its full colored bracket by the displayed expression, where $\sigma (L)$ is the signature of its linking matrix. The Kirby color handle-slide identity handles slides; $p_{+}{p}_{-}={\mathcal{D}}^2$ cancels disjoint plus- or minus-one-framed unknots. Kirby calculus therefore makes this an oriented 3-manifold invariant. This convention has ${\tau }_r(S^3)=1$.

// Target: module-theory.bigb

A Kirby diagram depicts a handle decomposition of a four-dimensional manifold using a framed link for the two-handles and dotted circles for the one-handles. The framing specifies the attaching normal directions; it controls the resulting intersection form and the boundary Dehn surgery.

If a meridian of a knot component in a surgery link links only one other component, filling it with coefficient $b\ne 0$ removes it and replaces the other coefficient $a$ by $a-1/b$. This gives the negative continued-fraction chain description of rational Dehn filling.