$F'=K'_1\cup \dots \cup K'_n$ are [*link homologous*]{} if there is a compact oriented $3$-manifold $W$ properly embedded in ${\bf R}^4
\times [0,1]$ such that $W$ has $n$ components $W_1, \dots, W_n$ with $\partial W_i= K_i \times \{0\} \cup (-K'_i)\times
\{1\}$. This relation is sometimes called [*link-cobordism*]{}, but that term also denotes the concordance relation. Since link homotopy implies link homology, the following theorem implies that triple linking invariants are link homotopy invariants (this fact is also seen from the sixth definition of ${\mbox{\rm Tlk}}$). For related topics, refer to [@Tim1; @TO; @Kirk; @KK; @Koschorke; @MassayRolf; @Ruberman; @NSato].

Triple linking invariants are link homology invariants: If $F=K_1\cup K_2\cup K_3$ and $F'=K'_1\cup K'_2\cup K'_3$ are link homologous, then ${\mbox{\rm Tlk}}(K_i,K_j,K_k)={\mbox{\rm Tlk}}(K'_i,K'_j,K'_k)$.

See Remark 6.2 for further information about link homology.

By Theorem 1.2, for any 3-component surface-link $F=K_1\cup K_2\cup K_3$, there exists a pair of integers $a$ and $b$ such that $$(*) \left\{
\begin{array}{l}
{\mbox{\rm Tlk}}(K_1,K_2,K_3)=-{\mbox{\rm Tlk}}(K_3,K_2,K_1)=-(a+b),\\
{\mbox{\rm Tlk}}(K_2,K_3,K_1)=-{\mbox{\rm Tlk}}(K_1,K_3,K_2)=b, \\
{\mbox{\rm Tlk}}(K_3,K_1,K_2)=-{\mbox{\rm Tlk}}(K_2,K_1,K_3)=a.
\end{array} \right.$$ In [@CJKLS], it is shown that for any pair of integers $a$ and $b$, there exists a surface-link $F$ whose triple linking numbers satisfy the above equations. However, that paper does not treat any problem about 