Kirby moves on integral framed links

Assume the Axiom of Choice. For an integral framed link diagram, Kirby move equivalence is the equivalence relation generated by the following operations.

1. Apply an orientation-preserving ambient isotopy to the framed link in $S^3$.

2. Handle slide: choose distinct components $K_i,K_j$, a parallel $K_j^{+}$ determined by the framing of $K_j$, and an embedded band whose ends attach to $K_i$ and $K_j^{+}$ and whose interior misses every link component. Replace $K_i$ by the band sum of $K_i$ with $K_j^{+}$ and keep the framing obtained by carrying the normal framing across the band. Orient the new component by the band-sum orientation.

3. Stabilization or destabilization: add or remove an unknot with framing $+1$ or $-1$ that lies in a $3$-ball disjoint from all the other components.

The integer framing of a component is measured from its preferred longitude as in the integral surgery definition. A planar link projection is not part of the move data; any drawing below is schematic.

Integral framing and surgery convention

Oriented links and ambient isotopy

Saveliev, Lecture 3 §3.2, Kirby moves K1 and K2

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

The left schematic shows two framed components and a band; the right schematic shows the new component as a band sum with the framed parallel of the second component. The circles do not encode crossing data or the embedding of the band in $S^3$; the Definition specifies those data explicitly.

\begin{tikzpicture}[>=stealth]
  \node[draw,circle,minimum size=0.72cm] (a) at (0,0) {$n_i$};
  \node[draw,circle,minimum size=0.72cm] (b) at (1.7,0) {$n_j$};
  \draw[thick] (0.35,0.15) .. controls (0.7,0.9) and (1.0,0.9) .. (1.35,0.15);
  \node at (0.85,0.68) {band};
  \draw[->,thick] (2.4,0) -- (3.25,0);
  \node[draw,circle,minimum size=0.72cm] (c) at (4.1,0) {$n_i'$};
  \node at (4.1,-0.72) {$K_i\mathbin{\#_b}K_j^+$};
\end{tikzpicture}