Lickorish–Wallace framed-link presentation theorem

Apply [F1] with the standard oriented $N=S^3$, target $M$, and empty external boundary. The hypotheses hold: both manifolds are connected, nonempty, compact and oriented, and their boundaries have the unique empty identification. The supplier gives a finite framed link $L\subset S^3$ with disjoint actual attaching tubes and a diffeomorphism $f:Q\to M$ from its simultaneous framed surgery. Give $Q$ the orientation extending the standard orientation on the link exterior; the orientation sign of $f$ remains to be checked. Its normal disk framings are the actual attaching framings of its four-dimensional two-handle trace; no arbitrary rational filling has been introduced.

Orient each of the finitely many components. In a component's oriented framed tube $S^1\times D^2$, the framed parallel is $t\mapsto (t,v)$ for a fixed nonzero normal vector $v$. Its projection to the core has degree $+1$. In the positive-meridian/preferred-longitude basis from [F2], the meridian has degree zero and the preferred longitude has degree one, so this parallel has class $n_i[{\mu }_i]+[{\lambda }_i]$ for an integer $n_i$. This also follows by writing the normal vector in the preferred trivialization: its winding is the integer $n_i$. The surgery meridian of a framed two-handle is precisely this parallel, by the explicit exchanged disk-factor construction in [F1]. Thus every component has exactly the required integer slope, with the full original normal framing retained.

The disjoint attaching tubes in step 1.1 permit these fillings simultaneously; step 2.1 identifies each with the integral convention in [F2]. If $f:Q\to M$ preserves orientation, this is the required presentation. Otherwise reflect the standard $S^3$ by an orientation-reversing diffeomorphism $r$, transport each component orientation, and replace each integer $n_i$ by $-n_i$. The positive meridian and preferred longitude of the reflected component satisfy $[{\mu }_i^{\mathrm{\prime }}]=-r_{\ast }[{\mu }_i]$ and $[{\lambda }_i^{\mathrm{\prime }}]=r_{\ast }[{\lambda }_i]$: reflection reverses the oriented normal disk, while preserving the degree-one and null-homology properties that characterize the longitude. Thus $r_{\ast }(n_i[{\mu }_i]+[{\lambda }_i])=-n_i[{\mu }_i^{\mathrm{\prime }}]+[{\lambda }_i^{\mathrm{\prime }}]$. To see the oriented effect on the filled manifold, write an original attaching map as $g_i:\mathrm{\partial }(D^2\times S^1)\to \mathrm{\partial }E$. On the filling solid torus use the orientation-reversing diffeomorphism $c(z,w)=(z,\bar{w})$, which fixes its oriented meridian. Choose reflected attaching maps $g_i^{\mathrm{\prime }}=r\circ g_i\circ c^{-1}$; these have the required slopes and glue oriented solid tori to the reflected exterior. The maps $r$ on the exterior and $c$ on each filling agree across these seams and their product collars, giving an orientation-reversing diffeomorphism $R:Q\to Q^{\mathrm{\prime }}$ to the reflected surgery manifold. Consequently $f\circ R^{-1}:Q^{\mathrm{\prime }}\to M$ preserves orientation. This covers both orientation signs without strengthening [F1].

If the link is empty, the same reflection argument applies directly to $S^3$. By [F2] its surgery result is $S^3$, and the empty diagram is an allowed presentation of the standard oriented $S^3$. Together with step 3.1 this proves the claim.