M_1|_{\partial N_2} \cdot
M_2|_{\partial N_2} \cdot
M_3|_{\partial N_2}) \\
&=
<u_1|_{\partial N_2} \cup
u_2|_{\partial N_2} \cup
u_3|_{\partial N_2}, [\partial N_2] >.
\end{array}$$

${\mbox{\rm Tlk}}_4(K_1,K_2,K_3)={\mbox{\rm Tlk}}_5(K_1,K_2,K_3).$

[*Proof.*]{} Let $i : \partial N_2 \to N_2$ be the inclusion map. In $H_0(N_2)$, we have $$\begin{array}{ll}
i_*(
M_1|_{\partial N_2} \cdot
M_2|_{\partial N_2} \cdot
M_3|_{\partial N_2})
&= -
i_*(
M_1|_{\partial N_2} \cdot
M_3|_{\partial N_2} \cdot
M_2|_{\partial N_2}) \\
&= -
i_*(
\partial_*(M_1|_{(N_2,\partial N_2)}) \cdot
\partial_*(M_3|_{(N_2,\partial N_2)}) \cdot
M_2|_{\partial N_2}) \\
&= -
i_*(
\partial_*(M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)}) \cdot
M_2|_{\partial N_2}) \\
&= -
(M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)}) \cdot
i_*(M_2|_{\partial N_2}) \\
&= -
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
(-[K_2]_{N_2}) \\
&=
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2}.
\end{array}$$ Thus $${\mbox{\rm Tlk}}_5(K_1,K_2,K_3) = \varepsilon_{N_2}(
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2}
).$$ It is obvious that $$\varepsilon_{N_2}(
M_1|_{(N_2,\partial N_2)} \cdot
M_3|_{(N_2,\partial N_2)} \cdot
[K_2]_{N_2} )
=
\varepsilon_{E_{13}}(
M_1|_{(E_{13},\partial E_{13})} \cdot
M_3|_{(E_{13},\partial E_{13})} \cdot
[K_2]_{E_{13}} )$$ and hence we have the result.  

${\rm Tlk}_6(K_1,K_2,K_3) =
\pm {\rm Tlk}_1(K_1,K_2,K_3).$

[*Proof.*]{} Since ${\rm Tlk}_6$ is an ambient isotopy invariant, we may assume that the surface-link $F= f(F_1) \cup f(F_2) \cup
f(F_3)$ is in general position with respect to the projection $p : {\bf R}^4 \to 