Integral framed-link surgery diagram

Assume the Axiom of Choice. It supplies countable choice for the chosen tubular data: for any sequence of nonempty sets, apply full choice to its range and compose with the index map. Let $L={\bigcup }_{i\in I}{K}_i$ be a finite oriented link in the standard oriented $3$-sphere, where $I$ is a finite index set. For each component $K_i$, choose a closed tubular neighbourhood $N(K_i)$, with the neighbourhoods pairwise disjoint, and include its exterior $E_{K_i}=S^3\setminus \mathrm{int}N(K_i)$, the positive meridian ${\mu }_i$, and the preferred longitude ${\lambda }_i$; the longitude is the degree-one parallel class in $H_1(\mathrm{\partial }{E}_{K_i};\mathbb{Z})$ that maps to zero in $H_1(E_{K_i};\mathbb{Z})$. For each $i\in I$, choose an integer $n_i$. The resulting integral framed-link surgery diagram is this link and integer family, with the surgery result defined by removing the interiors of all $N(K_i)$ and gluing an oriented solid torus to each boundary torus so that its meridian maps to $n_i[{\mu }_i]+[{\lambda }_i]$. If $I=\mathrm{\varnothing }$, the surgery result is $S^3$. A diagram is considered up to orientation-preserving ambient isotopy carrying each component framing; a planar projection is only a way of drawing the link.

Finite smooth oriented link and ambient isotopy

Chosen component exterior

Positive meridian and preferred longitude

Existence and uniqueness of each preferred longitude

Integer filling slope convention

Saveliev, Lecture 2 §2.2, framed links and integral surgery

Kirby, A Calculus for Framed Links in S3, §1