Boundary linking is intersection of bounding chains in the four-ball

Finite-chain model and independence. Write each rational chain as a finite rational sum of parametrized singular simplices. Within each labeled intersection factor, take the finite family of its simplices and iterated faces, identifying equal parametrized faces so that cancellations remain exact. Never identify simplices across the two factors merely because their maps agree: their perturbations must be independent. Preserve the given smooth product end collars and truncate at a smaller positive collar level. The remaining target is the boundaryless interior $B^{4\circ }$. Apply [F7] inductively over the finite face family, extending compatible face homotopies and fixing prescribed smooth faces. This gives a smooth chain replacement respecting the boundary operator, with the same rational coefficients; the homotopy prisms give relative homology. Applying [F7] with any prescribed open neighborhood as target keeps the replacement there. For general position, keep the two labeled families separate and make each pair of their open face strata transverse to the diagonal in $B^{4\circ }\times B^{4\circ }$. Choose faces in increasing dimension, fixing their previously treated subfaces. In Euclidean coordinates a cutoff-supported translation of the unfixed map is submersive in its parameter where the cutoff is positive; near the fixed part the pair is already transverse. The relative and parametric clauses of [F6] therefore give the required perturbation. Extend a chosen face homotopy consistently to all incident simplices in that factor, but not to the other factor. Finitely many face-pair conditions can be imposed simultaneously. Subdivision into finitely many compact coordinate pieces and sufficiently small parameters keep all changes inside prescribed open neighborhoods and away from the end collars. For complementary two-chains the diagonal preimages are finite signed sets; lower-dimensional face pairs are empty. Define the intersection by summing these signs with the rational product coefficients. This agrees with the given transverse interior count. Now fix $Q$ and two choices $P,P^{\mathrm{\prime }}$ bounding $\alpha$, using facewise-transverse representatives with their identical end collars. Their difference $C=P-P^{\mathrm{\prime }}$ is a smooth rational two-cycle compactly supported in $B^{4\circ }$. By [F3], choose a finite continuous rational three-chain $Z$ there with $\mathrm{\partial }Z=C$. Smooth $Z$ relative to its prescribed boundary $C$ by [F7], identifying faces within the $Z$ family only. Keeping $Q$ independent and fixed, perturb the remaining $Z$ faces relative to $C$, whose intersections with $Q$ are already transverse. Each simplex-pair diagonal preimage is a compact oriented one-manifold; codimension-one faces give its endpoints, while codimension-two faces have negative expected dimension. The local half-space model and [F4] give the boundary signs. Sum with the rational product coefficients: internal face endpoints cancel because the same face map occurs with opposite boundary coefficients. What remains is $(\mathrm{\partial }Z)\cdot Q=C\cdot Q$, together with the term from $Z$ and $\mathrm{\partial }Q$. That latter term is empty since $Z$ is compactly supported in the interior and $\mathrm{\partial }Q=\beta \subset S^3$. By [F2] the total signed boundary count is zero, so $P\cdot Q=P^{\mathrm{\prime }}\cdot Q$. Apply the same finite construction to homotopy prisms to obtain independence of the replacements. Interchanging the two two-dimensional factors introduces no sign and proves independence of the second chain. All operations are rational-linear, giving the same assertions for finite rational combinations. No embedded-manifold realization of $Z$ is used.

By [F5], choose an oriented Seifert surface $S_{\beta }\subset S^3$ for each knot component of $\beta$, and similarly choose a cap for each component of $\alpha$. It suffices by bilinearity to consider one knot component of each. Choose a collar depth $0<ϵ$ in $B^4$. Make $Q$ a graph of $S_{\beta }$ in the boundary collar with depth zero on $\beta$ and strictly between $0$ and $ϵ\mathrm{/}2$ on its interior. To give this graph a literal product end collar, choose $0<\delta <ϵ\mathrm{/}2$, attach $\beta \times [0,\delta ]$, and join its inner edge to the graph inside a tubular neighborhood of $\beta$ disjoint from $\alpha$, leaving the graph unchanged outside that neighborhood; choose the join below depth $ϵ\mathrm{/}2$. This adds no intersections with $P$ and changes none of the signs computed below. Make $P$ consist of the product collar $\alpha \times [0,ϵ]$ followed by a cap at depth $ϵ$. Then $P$ and $Q$ intersect exactly above $\alpha \cap S_{\beta }$. At each such point, the oriented tangent basis of the collar of $P$ is $(n_{\mathrm{o}\mathrm{u}\mathrm{t}},a)$, where $a$ is the positive tangent to $\alpha$ and $n_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the outward normal of $B^4$; the tangent basis of $Q$ is a positive basis $(s_1,s_2)$ of $S_{\beta }$, up to radial components which do not change the determinant. By [F4], $(n_{\mathrm{o}\mathrm{u}\mathrm{t}},a,s_1,s_2)$ is positive exactly when $(a,s_1,s_2)$ is positive in the boundary orientation. Thus the $4$-dimensional intersection sign is exactly the $3$-dimensional sign of $\alpha \cap S_{\beta }$. Hence $P\cdot Q=\mathrm{lk}(\alpha ,\beta )$ by [F1].

Step 1.1 shows the intersection number is independent of admissible collared bounding chains, and step 1.2 computes it for one convenient choice. Hence the equality holds for all such $P,Q$ when the boundaries are knots. Expanding finite rational cycles into their oriented knot components and using bilinearity of both signed intersection and linking proves the stated equality.