$E_3$. This is equal to the image of $M_1|_{K_2} \in H_1(K_2)$ under the inclusion-induced homomorphism $H_1(K_2) \to H_1(E_3)$ and also equal to the image of $M_1|_{(E_{13}, \partial E_{13})}
\cdot  [K_2]_{E_{13}} \in H_1(E_{13})$ under the inclusion-induced homomorphism $H_1(E_{13}) \to H_1(E_3)$, where $[K_2]_{E_{13}} \in H_2(E_{13})$ is represented by $K_2$. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_3(K_1, K_2, K_3)
&=
{\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) \\
&=
\varepsilon_{E_3} ([M_1 \cap K_2]_{E_3} \cdot M_3),
\end{array}$$ where $M_3 \in H_3(E_3, \partial E_3)$ and $\varepsilon_{E_3}: H_0(E_3) \to {\bf Z}$ is the augmentation.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)=-{\mbox{\rm Tlk}}_3(K_1,K_2,K_3).$

[*Proof.*]{} In the situation of the proof of Lemma 2.2, $[M_1 \cap K_2]_{E_3} \in H_1(E_3)$ is represented by the $1$-cycle $-D_{12}^{\ell}$. Therefore, by Lemma 2.2, we have $$\begin{array}{ll}
{\mbox{\rm Tlk}}_3(K_1,K_2,K_3)
&= {\mbox{\rm Link}}([M_1 \cap K_2]_{E_3}, K_3) \\
&= {\mbox{\rm Link}}(-D_{12}^{\ell}, K_3) \\
&= -{\mbox{\rm Tlk}}_1(K_1,K_2,K_3). \ {\hbox{\rule{6pt}{6pt}}}\end{array}$$

We denote by $[M_1 \cap M_3]_{(E_{13}, \partial E_{13})}
\in H_1(E_{13}, \partial E_{13})$ the class of the intersection $M_1 \cap M_2$ as a $2$-cycle in $(E_{13}, \partial E_{13})$ when we regard $M_i$ as a $3$-chain. This is equal to the intersection product $M_1 |_{(E_{13}, \partial E_{13})}
\cdot M_3 |_{(E_{13}, \partial E_{13})}
\in H_2(E_{13}, \partial E_{13})$. Let $$\begin{array}{ll}
{\mbox{\rm Tlk}}_4(K_1, K_2, K_3)
&=
\varepsilon_{E_{13}} ([M_1 \cap M_3]_{(E_{13}, \partial E_{13})}
  