Framed link

A framed link is a link together with a homotopy class of nonzero normal vector field on every component. Relative to the Seifert framing, each component framing is encoded by an integer.

The Seifert framing pushes a knot in the direction tangent to a Seifert surface and normal to its boundary. The knot and its push-off then have linking number zero.

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 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.

Kirby calculus changes framed-link diagrams by handle slides and creation or cancellation of standard handle pairs without changing the represented four-manifold up to diffeomorphism.

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.