(M_2|_{\partial E_3})_{\partial E} \cdot
(M_3|_{\partial E_3})_{\partial E} \cdot
(M_1|_{\partial E_3})_{\partial E} \\
& +
(M_3|_{\partial E_1})_{\partial E} \cdot
(M_1|_{\partial E_1})_{\partial E} \cdot
(M_2|_{\partial E_1})_{\partial E}
) \\
=
& - \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} \\
& +
(M_1|_{\partial E_3})_{\partial E} \cdot
(M_2|_{\partial E_3})_{\partial E} \cdot
(M_3|_{\partial E_3})_{\partial E} \\
& +
(M_1|_{\partial E_1})_{\partial E} \cdot
(M_2|_{\partial E_1})_{\partial E} \cdot
(M_3|_{\partial E_1})_{\partial E}
) \\
=
& - \varepsilon_{\partial E}(
M_1|_{\partial E} \cdot
M_2|_{\partial E} \cdot
M_3|_{\partial E}) \\
=
& - \varepsilon_{\partial E}(
\partial_*(M_1|_{(E,\partial E)}) \cdot
\partial_*(M_2|_{(E,\partial E)}) \cdot
\partial_*(M_3|_{(E,\partial E)})) \\
=
& - \varepsilon_{\partial E}( \partial_* (
M_1|_{(E,\partial E)} \cdot
M_2|_{(E,\partial E)} \cdot
M_3|_{(E,\partial E)})) \\
=
& 0. \  {\hbox{\rule{6pt}{6pt}}}\end{array}$$ [*Proof of Theorem $1.3$*]{} (i) The intersection number between two oriented curves on a 2-sphere vanishes. By Lemma 2.1, we have ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

\(ii) This is an immediate consequence of (i) and Theorem 1.2(ii).  ${\hbox{\rule{6pt}{6pt}}}$

[*Proof of Theorem $1.4$*]{} If ${\mbox{\rm Alk}}(K_2, K_3)=0$, then ${\mbox{\rm Tlk}}_3(K_1, K_2, K_3) ={\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) = 0$, for $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ is the image of $M_1|_{K_2} \in H_1(K_2)=0$. By Theorem 1.2, we have ${\mbox{\rm Tlk}}(K_3, K_2, K_1)=0$.  ${\hbox{\rule{6pt}{6pt}}}$

