all the triple points of type $(i,j,k)$ on a surface diagram of $F$;

$${\mbox{\rm Tlk}}(K_i,K_j,K_k)=
\sum_{{\scriptstyle \tau:\
\mbox{\footnotesize type}\ (i,j,k)}}
\ \varepsilon(\tau).$$ It is proved in [@CJKLS] that this number is an invariant of the surface-link $F$ (independent of a diagram in ${\bf R}^3$) by use of Roseman moves (Reidemeister moves for surface-link diagrams) [@Rose], and that this invariant vanishes in the case that $i=k$; that is, ${\mbox{\rm Tlk}}(K_i,K_j,K_i)=0$ for $i\neq j$. Hence throughout this paper, we always assume that $i, j, k$ are all distinct whenever we refer to ${\mbox{\rm Tlk}}(K_i,K_j,K_k)={\mbox{\rm Tlk}}_1(K_i,K_j,K_k)$.

We prove the following properties of triple linking by using the above interpretations.

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

-   ${\mbox{\rm Tlk}}(K_1,K_2,K_3)+{\mbox{\rm Tlk}}(K_2,K_3,K_1)+{\mbox{\rm Tlk}}(K_3,K_1,K_2)=0$.

<!-- -->

-   If $K_2$ is homeomorphic to a $2$-sphere, then ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

-   If both of $K_1$ and $K_3$ are homeomorphic to a $2$-sphere, then ${\mbox{\rm Tlk}}(K_1,K_2,K_3)=0$.

In [@NSato] the asymmetric linking number ${\mbox{\rm Alk}}(K,K')$ for a two component oriented surface-link $F=K \cup K'$ was defined to be the non-negative generator of the image of $H_1(K) \rightarrow H_1(S^4 \backslash K') \cong {\bf Z}$.

If ${\mbox{\rm Alk}}(K_2, K_3)=0$, then ${\mbox{\rm Tlk}}(K_1, K_2,K_3)= {\mbox{\rm Tlk}}(K_3, K_2, K_1)= 0$.

Two $n$-component surface-links $F=K_1\cup \dots \cup K_n$ and 