\backslash (N_1\cup N_2\cup N_3)),$$ where Cl denotes the closure. We denote by $M_i$ a $3$-chain in $S^4$ with $\partial M_i = K_i$ for $i=1,2,3$ (the reader may suppose that it is a Seifert hypersurface for $K_i$, i.e., a compact oriented $3$-manifold embedded in $S^4$ with $\partial M_i=K_i$). We also denote by $M_i$ the homology class in $$H_3(S^4,K_i) {\cong} H_3(S^4,N_i) {\cong}
H_3(E_i,\partial E_i)$$ represented by $M_i$. By $u_i \in H^1(E_i)$ we denote the Poincaré dual of $M_i \in H_3(E_i,\partial E_i)$, i.e., $M_i= P(u_i) = u_i\cap[E_i]$. For a subset $X$ of $E_i$, we will denote by $u_i|_{X} \in H^1(X)$ the image of $u_i$ by the inclusion-induced homomorphism $H^1(E_i) \to H^1(X)$. Moreover, if $X$ is an $n$-manifold, we denote by $M_i|_{(X, \partial X)}$ (or $M_i|_{X}$ if $\partial X =\emptyset$) the Poincaré dual $P_X(u_i|_{X})= (u_i|_{X}) \cap [X] \in
H_{n-1}(X, \partial X)$ of $u_i|X$.

For $i \in \{1,3\}$, since $K_2 \subset E_i$, $M_i|_{K_2} \in H_1(K_2)$ is defined. (When we consider $M_i$ as a $3$-chain, the intersection of $M_i$ and $K_2$ (as a $1$-cycle in $K_2$) represents $M_i|_{K_2}$.) Let $${\mbox{\rm Tlk}}_2(K_1, K_2, K_3)=
\varepsilon_{K_2}(M_1|_{K_2} \cdot M_3|_{K_2}),$$ where $\varepsilon_{K_2}: H_0(K_2) \to {\bf Z}$ is the augmentation.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)={\mbox{\rm Tlk}}_2(K_1,K_2,K_3).$

[*Proof.*]{} We assume $F \subset {\bf R}^4 \subset 