S^4$ and continue the situation of the proof of Lemma 2.2. Let $M'_1$ be a 3-chain in ${\bf R}^4 (\subset S^4)$ with $\partial M'_1= K_1$ and consider a $3$-chain $M_1$ such that $$M_1 = -\cup_{\lambda \in [-R, 0]} t_\lambda(K_1)
+ t_{-R}(M'_1)$$ with $\partial M_1= K_1$. The intersection of $M_1$ and $K_2$ is equal to that of $-\cup_{\lambda \in [-R, 0]} t_\lambda(K_1)$ and $K_2$ which is the 1-chain $-D_{12}^{\ell}$ in $K_2$, and the intersection of $M_3$ and $K_2$ is equal to that of $\cup_{\lambda \in [0, R]} t_\lambda(K_3)$ and $K_2$ which is the 1-chain $D_{23}^u$ in $K_2$. Therefore, by Lemma 2.1, we have $$\begin{array}{ll}
{\mbox{\rm Tlk}}_1(K_1,K_2,K_3)
&= -{\mbox{\rm Int}}_{K_2}(D_{12}^{\ell},D_{23}^u) \\
&= -\varepsilon_{K_2}(D_{12}^{\ell} \cdot D_{23}^u) \\
&= -\varepsilon_{K_2}(-M_1|_{K_2} \cdot M_3|_{K_2})\\
&= {\mbox{\rm Tlk}}_2(K_1,K_2,K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

[**Remark.**]{} The argument in Lemmas 2.2 and 3.1 implies that for any Seifert hypersurface $M_1$ for $K_1$, the intersection of $M_1$ and $K_2$ (as a 1-cycle in $K_2$) is homologous to $-D_{12}^{\ell}$, and that for any Seifert hypersurface $M_3$ for $K_3$, the intersection of $M_3$ and $K_2$ (as a 1-cycle in $K_2$) is homologous to $D_{23}^u$.

We denote by $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ the homology class of the intersection $M_1 \cap K_2$ as a $1$-cycle in 