\cdot [K_2]_{E_{13}}) \\
&=
\varepsilon_{E_{13}} (M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})} \cdot [K_2]_{E_{13}}).
\end{array}$$

${\mbox{\rm Tlk}}_3(K_1,K_2,K_3)={\mbox{\rm Tlk}}_4(K_1,K_2,K_3).$

[*Proof.*]{} Let $i_*: H_*(E_{13}) \to H_*(E_3)$ and $i^*: H^*(E_3) \to H^*(E_{13})$ be the inclusion-induced homomorphisms. Recall that $[M_1 \cap K_2]_{E_3}= i_*(M_1|_{(E_{13}, \partial E_{13})}
\cdot  [K_2]_{E_{13}})$. Thus,

$$\begin{array}{ll}
{\mbox{\rm Tlk}}_4(K_1, K_2, K_3)
&=
\varepsilon_{E_{13}} (
M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) \\
&=
\varepsilon_{E_{3}} \circ i_* (
M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}})   \\
&= -
\varepsilon_{E_{3}} \circ i_* (
M_3 |_{(E_{13}, \partial E_{13})}
\cdot M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}})  \\
&= -
\varepsilon_{E_{3}} \circ i_* (
u_3 |_{E_{13}}
\cap (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}} \circ i_* (
i^*(u_3)
\cap (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}}  (
u_3
\cap i_* (M_1 |_{(E_{13}, \partial E_{13})}
\cdot [K_2]_{E_{13}}) )  \\
&= -
\varepsilon_{E_{3}}  (
M_3
\cdot [M_1 \cap K_2]_{E_3} )  \\
&=
\varepsilon_{E_{3}}  (
[M_1 \cap K_2]_{E_3} \cdot M_3
)  \\
&=
{\mbox{\rm Tlk}}_3(K_1, K_2, K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

For $i \in \{1,2,3\}$, since $\partial N_2 \subset E_i$, $M_i|_{\partial N_2} \in H_2(\partial N_2)$ is defined. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_5(K_1,K_2,K_3)
&=
\varepsilon_{\partial N_2}(
