Alexander polynomial one implies topological sliceness

Apply [F1] to obtain the actual surface $\mathrm{\Sigma }$, complement $N$, short curves $s_1,\dots ,s_g$, marked annuli and smooth disk/dual datum. By [A1] and [F2], [F3] applies with $V=N$. Its hypothesis concerns smooth maps and smooth collars; those are explicitly supplied here. Thus the compression disks have disjoint embedded standard product handle charts with the prescribed attaching framings. If $g=0$, the original $\mathrm{\Sigma }$ is already the required disk and the following finite surgeries are empty.

Join each produced disk to its short curve on $\mathrm{\Sigma }$ along the marked negative-normal annulus. These annuli are disjoint, and their interiors lie on the $\mathrm{\Sigma }$ side of $\mathrm{\partial }N$; the disks lie on its $N$ side. The supplied collars identify their product charts at the seam. The resulting joined disks therefore have product normal neighborhoods whose first boundary normal direction is the normal direction of $s_i$ within $\mathrm{\Sigma }$. Choose mutually disjoint sufficiently small product neighborhoods. Replace the annulus $s_i\times (-\eta ,\eta )$ in $\mathrm{\Sigma }$ by the two disk parallels at $\pm \eta$ in that first normal direction, with opposite orientations. The attaching collars give two coordinate half-planes meeting on their common boundary line. Rounding their quadrant in those product coordinates yields a locally flat surface. This uses the product charts actually supplied by [F3], rather than a normal-neighborhood theorem for an arbitrary TOP surface. All changes fix the outer knot collar.

Cutting along the geometric short curves leaves a connected surface. Indeed, if its complementary-component graph had more than one vertex, choose a proper nonempty vertex subset with an edge to its complement. The oriented boundary of the corresponding union gives a nonzero signed relation among the independent classes $[s_i]$: every crossing edge occurs once, while edges internal to the union cancel. This is impossible. Hence capping these cut curves leaves the result connected. Removing the annuli changes no Euler characteristic, while the $2g$ disk caps raise it from $1-2g$ to $1$. The connected oriented one-boundary surface is a disk: the geometric cut-system polygon is a sphere with $2g+1$ boundary circles, and the caps fill its $2g$ inner circles. Its only outer boundary is still $K$, and its original boundary product collar is unchanged. Finally radial rescaling of the literal radius-$3\mathrm{/}2$ ball to $B^4$ carries this locally flat proper disk to the desired one.