Small framed push-offs define a well-defined self-linking number

Choose a tubular chart $\mathrm{\Phi }:\mathrm{\Omega }\to U$ as in [F1]. The graph map $x\mapsto (x,\epsilon s(x))$ is an embedding because its projection to the base $K$ is the identity; it is disjoint from the zero section because $s(x)\mathrm{\ne }0$. Since $\mathrm{\Phi }$ is a diffeomorphism on $\mathrm{\Omega }$, its image $K_{\epsilon s}$ is embedded and disjoint from $K$ whenever the graph is contained in $\mathrm{\Omega }$. The zero section is compact, so an open neighbourhood of it contains all sufficiently small scalar multiples of $s$ over $K$; hence such $\epsilon$ exist. The orientation of $K_{\epsilon s}$ is transported from the base orientation by its projection to $K$.

Fix $a\in (0,1\mathrm{/}2)$ and use the smooth plateau ${\chi }_a$ of [F6]. For two sufficiently small radii ${\epsilon }_0,{\epsilon }_1>0$ in the same chart, define $\epsilon (u)=(1-{\chi }_a(u)){\epsilon }_0+{\chi }_a(u){\epsilon }_1$. The family of embeddings of $P:=S^1\bigsqcup S^1$ whose first component is $K$ and whose second is $K_{\epsilon (u)s}$ is smooth, stays embedded and disjoint by step 1.1, and is constant on endpoint collars. Here $P$ is compact and boundaryless, so [F4] extends this family to an ambient isotopy; [F5] shows that the linking number is unchanged. If $s_v$ is a smooth homotopy through nonzero sections, compactness of $K\times [0,1]$ gives one small scalar $\epsilon$ for which all graphs lie in $\mathrm{\Omega }$. Replacing $v$ by ${\chi }_a(u)$ in the family $K\cup K_{\epsilon s_v}$ again gives a smooth family of embeddings of the same compact boundaryless source, stationary near its endpoints; [F4] and [F5] show that the linking number is independent of the representative section.

Let ${\mathrm{\Phi }}_0:{\mathrm{\Omega }}_0\to S^3$ and ${\mathrm{\Phi }}_1:{\mathrm{\Omega }}_1\to S^3$ be two normalized tubular charts. By [F2], choose a smooth bundle metric on $\nu K$. Since $K$ is compact and ${\mathrm{\Omega }}_0\cap {\mathrm{\Omega }}_1$ is an open neighbourhood of the zero section, finitely many bundle trivializations give a common radius $r>0$ such that the open metric disk bundle ${\mathrm{\Omega }}_r=\{v\in \nu K:\parallel v\parallel <r\}$ is contained in both domains. Restrict both charts to this same ${\mathrm{\Omega }}_r$. They now satisfy the common-domain hypothesis of [F3]. After its permitted shrinking, [F3] gives an ambient isotopy $H_t$ with $H_1\circ {\mathrm{\Phi }}_0={\mathrm{\Phi }}_1$ on a neighbourhood of the compact zero section. Choose $\epsilon >0$ small enough that the graph of $\epsilon s$ lies in this neighbourhood. Then $H_1$ carries the oriented pair formed from $K$ and its ${\mathrm{\Phi }}_0$-push-off to the corresponding pair for ${\mathrm{\Phi }}_1$, so [F5] makes their linking numbers equal. Steps 1.1–2.1 prove independence of chart, sufficiently small radius and representative section. Countable Choice is used through the metric and tubular-neighbourhood suppliers; the finite linking computations are choice-free.