\\
&= -
M_1|_{\partial E_2} \cdot M_2|_{\partial E_2} \cdot
M_3|_{\partial E_2}.   \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

[*Proof of Theorem $1.2$*]{} (i) $$\begin{array}{ll}
{\mbox{\rm Tlk}}_2(K_1, K_2, K_3)
&=
\varepsilon_{K_2}(M_1|_{K_2} \cdot M_3|_{K_2}) \\
&= -
\varepsilon_{K_2}(M_3|_{K_2} \cdot M_1|_{K_2}) \\
&= -
{\mbox{\rm Tlk}}_2(K_3, K_2, K_1).
\end{array}$$

\(ii) Note that $M_i|_{(E, \partial E)} = P_{E}(u_i|_{E})
\in H_3(E, \partial E)$ is the image of $M_i$ under $$H_3(E_i, \partial E_i) \to
H_3(E_i, \partial E_i \cup N_j \cup N_k)
\cong
H_3(E, \partial E),$$ and $M_i|_{\partial E_2}
\in H_2(\partial E_2)$ is the image of $M_i|_{(E, \partial E)}$ under $$H_3(E, \partial E) \to H_2(\partial E)
\cong
H_2(\partial E_1) \oplus H_2(\partial E_2) \oplus
H_2(\partial E_3) \to H_2(\partial E_2),$$ the boundary operator followed by the projection to $H_2(\partial E_2)$. We denote by $(M_i|_{\partial E_2})_{\partial E} \in H_2(\partial E)$ the image of $M_i|_{\partial E_2}$ under the inclusion-induced homomorphism $H_2(\partial E_2) \to H_2(\partial E)$. By Lemma 4.1, $${\mbox{\rm Tlk}}_5(K_1, K_2, K_3)= -
\varepsilon_{\partial E}(
(M_1|_{\partial E_2})_{\partial E} \cdot
(M_2|_{\partial E_2})_{\partial E} \cdot
(M_3|_{\partial E_2})_{\partial E}).$$ Thus, we have $$\begin{array} {rl}
{\mbox{\rm Tlk}}_5(K_1, K_2, K_3) + &
{\mbox{\rm Tlk}}_5(K_2, K_3, K_1) +
{\mbox{\rm Tlk}}_5(K_3, K_1, K_2) \\
=
& - \varepsilon_{\partial E}(
(M_1|_{\partial E_2})_{\partial E} \cdot
(M_2|_{\partial E_2})_{\partial E} \cdot
(M_3|_{\partial E_2})_{\partial E} \\
& +
