Puncturing a smooth four-ball surface gives a genus-preserving knot cobordism

Move a genuine ambient ball by translations. Choose an interior point $p$ of $F$ and a radius $r>0$ with $\mathrm{\mid }p\mathrm{\mid }+r<1$. The embeddings of the closed radius-$r$ ball $D$ centered at $p$ given by $x\mapsto x-tp$, $0\le t\le 1$, stay in the interior of $B^4$: the norm of a point in their image is at most $(1-t)\mathrm{\mid }p\mathrm{\mid }+r<1$. Their compact track has a neighborhood with closure in the interior of $B^4$. By [F1] this isotopy extends ambiently in ${\mathbb{R}}^4$ with support in that neighborhood. Its time-one map fixes an ambient neighborhood of the outer boundary, preserves orientation and takes $p$ to $0$, while agreeing with translation on the whole ball $D$. Replace $F$ by its image. No conclusion about any nonround ball complement has been drawn.

Flatten the surface on an actual neighborhood of zero. Let $P$ be the tangent plane of the new surface at $0$. Orthogonal projection to $P$ has invertible derivative on that tangent plane, so [F3] writes the surface as $(u,h(u))$ in $P\oplus P^{\perp }$, with $h(0)=0$ and $Dh(0)=0$. Choose one sufficiently small closed disk in the $u$ coordinates. Its graph is a compact embedded disk in the interior of $B^4$. The maps $u\mapsto (u,(1-t)h(u))$ are an isotopy of that compact disk, since projection to $u$ remains injective with injective differential. Choose the disk small enough that its entire compact track lies in the interior of $B^4$. Apply [F1] again there, obtaining an ambient orientation-preserving diffeomorphism that fixes an ambient neighborhood of the outer boundary and maps the full graph disk to the flat disk. The surface is now exactly $P$ near $0$: the disk interior is an open neighborhood of $0$ within the embedded surface. Its complement outside a smaller such graph disk is compact and misses $0$, so it has positive distance from $0$. Choose $\epsilon >0$ smaller than that distance and the flat disk radius, and less than $1$. Then the surface meets the actual round ball $B_{\epsilon }^4$ precisely in $P\cap B_{\epsilon }^4$. Compose the two ambient diffeomorphisms to obtain $\mathrm{\Psi }$. They fix a common ambient neighborhood $N$ of the outer boundary. For the supplied collar $c:K\times [0,\eta )\to F$, the open set $c^{-1}(N)$ contains $K\times \{0\}$. A finite cover of the compact circle by product neighborhoods gives one $0<\delta <\eta$ with $c(K\times [0,\delta ])\subset N$. Thus $\mathrm{\Psi }$ fixes this sufficiently small restriction of the supplied collar; no fixedness of its entire image is asserted.

Use only the concentric round shell. The explicit map $(z,t)\mapsto (\epsilon +(1-\epsilon )t)z$ is a diffeomorphism $S^3\times [0,1]\to B^4\setminus \mathrm{int}{B}_{\epsilon }^4$. The inner boundary curve is the great circle $P\cap S_{\epsilon }^3$, hence a standard unknot after radial rescaling; an orthonormal basis of $P$ extends by finite linear algebra to an oriented orthonormal basis of ${\mathbb{R}}^4$, giving an orientation-preserving linear rotation identifying it with the usual equator. Near the inner sphere the surface is a radial product in $P$, so this radial shell coordinate already supplies its product collar.

Respect a restriction of the given outer surface collar. For completeness the original outer surface collar need not be radial. In its smooth proper half-space chart the radial coordinate has nonzero inward derivative along a transverse collar direction. In a supplied parametrization of that collar, [F3] and compactness of its boundary circle therefore reparametrize it, for $r$ near $1$, as $(r{f}_r(\theta ))$ with $f_r:S^1\hookrightarrow S^3$ a smooth embedding family and $f_1$ the original parametrized $K$. By [F1] on $S^3$, choose a smooth ambient family $G_r$ with $G_1=\mathrm{i}\mathrm{d}$ and $G_r{f}_1=f_r$. Choose a smooth function $h(r)$ equal to $r$ near $1$ and to $1$ near the inner edge of this thin outer collar, with values in the parameter interval where $G$ is defined. The fiberwise diffeomorphism $(z,r)\mapsto (G_{h(r)}^{-1}z,r)$ is the identity at $r=1$ and near that inner edge, so extends as the identity over the rest of the shell; near the outer end it takes the surface to the literal product $K\times [1-\delta ,1]$. Thus adjust the shell identification by this map. It leaves the inner sphere and the boundary knot fixed, and changes neither genus nor the fixedness of a sufficiently small restriction of the supplied outer collar by $\mathrm{\Psi }$ from steps 1.1–2.1. Removing one interior disk from a connected surface leaves it connected, retains genus and adds one boundary circle; its Euler characteristic decreases from $1-2g$ to $-2g$. Orient the resulting shell surface as a cobordism from the inner unknot to $K$. Reversing the interval and the surface orientation gives the reverse direction if desired; a half-turn reverses the orientation of the standard unknot. This proves the full genus, orientation and collared-end assertions under countable choice, using only an explicitly concentric shell.