Dehn surgery on a framed link

Integral Dehn surgery removes a tubular neighborhood of every component and reglues a solid torus so that its meridian follows the slope specified by the framing.

The framing specifies the integer slope in this special case of Dehn surgery.

Relative to the meridian of a knot $\mu$ and Seifert longitude $\lambda$, an integer surgery coefficient $n$ means that the filling meridian of a solid torus is identified with $n\mu +\lambda$.

The surgery trace $W(L)$ is obtained from $B^4$ by attaching one two-handle along every component of a framed link $L\subset S^3$. Its boundary is the three-manifold produced by surgery on $L$.

The surgery linking matrix has the framing coefficients on its diagonal and pairwise linking numbers off the diagonal. It represents the intersection form on the second homology of the surgery trace, presents the first homology of its boundary, and has kernel isomorphic to the boundary's second homology.

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.

For coprime integers $p>0$ and $q$, the lens space $L(p,q)$ is the quotient of $S^3\subset {\mathbb{C}}^2$ by $(z_1,z_2)\mapsto (e^{2\pi i/p}{z}_1,e^{2\pi iq/p}{z}_2)$. Equivalently, it is obtained by an appropriate rational surgery on the unknot.

The quotient model of a lens space splits into two solid tori. In the boundary angle lattice ${\mathbb{Z}}^2+\mathbb{Z}(1/p,q/p)$, choose ${\mu }_1=(0,1)$ and ${\lambda }_1=(1/p,q/p)$. The other meridian is $(1,0)=-q{\mu }_1+p{\lambda }_1$. A second column completing this primitive slope to a determinant-one matrix specifies a gluing map after reversing one boundary-basis convention. Different completions differ by a meridional Dehn twist extending across the second solid torus, so they give the same three-manifold.

With a consistent surgery orientation convention, $L(p,q)$ and $L(p,q^{\prime })$ are orientation-preservingly related by a homeomorphism exactly when $q^{\prime }\equiv q^{\pm 1}(\mathrm{mod}p)$. An orientation-reversing homeomorphism instead requires $q^{\prime }\equiv -q^{\pm 1}(\mathrm{mod}p)$.