Kirby moves preserve the oriented surgery manifold

For an ambient isotopy, carry each component neighbourhood, meridian, and preferred longitude along the isotopy. The endpoint diffeomorphism of $S^3$ therefore maps every filling meridian to the same framed slope on the corresponding endpoint torus. Extend it over each attached solid torus by the product identification used in the filling. The resulting map is an orientation-preserving diffeomorphism of the filled manifolds.

Applying [A1] to the family of values of any sequence $(X_n)$ in [F8] gives a choice function on that range; composing it with $n\mapsto X_n$ supplies the choice function required by [F8]. Thus the ${\mathrm{A}\mathrm{C}}_{\omega }$ hypothesis of [F3]–[F6] holds. For a diagram $D$, form its $4$-dimensional surgery trace from $S^3\times [0,1]$ by attaching one $2$-handle for each framed link component. Orient the product by $dt$ followed by the fixed orientation of $S^3$, and orient each handle so its attaching identification reverses the induced boundary orientations of the handle and cylinder, as required by [F5]; this is possible because each attaching region $S^1\times D^2$ is connected. The trace is connected. On the outgoing face, the unaltered part $E_D=S^3\setminus {\bigcup }_i\mathrm{int}N(K_i)$ has the standard $S^3$ orientation from the upper product face, exactly the orientation that the filling conventions [F1, F7] extend over the attached solid tori. The link exterior is nonempty and connected (paths in $S^3$ can be perturbed off the one-dimensional link; for the empty diagram $E_D=S^3$). Thus the identity-on-$E_D$ identification in [F6] preserves the surgery orientation. The link slide of [F2] is exactly the $k=2$ handle slide of [F3], including its full transported normal framing. Since $1\le 2\le 4-2$, [F4] supplies a diffeomorphism between the two traces relative to the incoming $S^3$ face. Both traces have the product orientation there and the diffeomorphism fixes that face; its tangent derivative is the identity and its normal derivative preserves the interior side, so it preserves the trace orientation. Its restriction preserves the induced orientations on the outgoing faces. Composing with the identifications in [F6] gives the required orientation-preserving diffeomorphism of surgery results.

Suppose a split unknot is added with coefficient $\epsilon \in \{1,-1\}$ in a ball disjoint from the rest of the link. The separating sphere shows that the filled manifold is the connected sum of the old surgery result with the filling of the unknot exterior along $\epsilon [{\mu }_U]+[{\lambda }_U]$. For the standard unknot, take $E_U=\{\mathrm{\mid }{z}_2\mathrm{\mid }\ge 1\mathrm{/}\sqrt{2}\}\subset S^3\subset {\mathbb{C}}^2$, identified with $D^2\times S^1$ by $(z_1,z_2)\mapsto (z_1,z_2\mathrm{/}\mathrm{\mid }{z}_2\mathrm{\mid })$. Its preferred longitude ${\lambda }_U$ is the meridian of this solid torus, and ${\mu }_U$ is its core longitude. The diffeomorphism $g_{\epsilon }(z,w)=(w^{-\epsilon }z,w)$ of $D^2\times S^1$ preserves orientation, fixes $[{\lambda }_U]$, and sends $[{\mu }_U]$ to $[{\mu }_U]-\epsilon [{\lambda }_U]$; hence it sends the slope $\epsilon [{\mu }_U]+[{\lambda }_U]$ to the unoriented meridional slope $[{\mu }_U]$. Choose the $S^3$ gluing with meridian $[{\mu }_U]$ and the $\epsilon$-filling gluing as its pullback by $g_{\epsilon }^{-1}$. The map $g_{\epsilon }$ on the exterior and the identity on the attached solid torus then give an orientation-preserving diffeomorphism between the two fillings. Filling along $[{\mu }_U]$ restores the original unknot neighbourhood, so the $\epsilon$-framed unknot filling is orientation-preservingly diffeomorphic to $S^3$. Hence the connected sum is the original manifold: deleting a ball and regluing the complementary ball of $S^3$ restores the same manifold. Removing a split $\epsilon$-framed unknot is the inverse operation.

Each allowed move preserves the oriented surgery manifold by steps 1.1–1.3. Composing the finitely many orientation-preserving diffeomorphisms along a move sequence proves the assertion for Kirby-move-equivalent diagrams.