notion of ${\mbox{\rm Tlk}}_1$ to a diagram in ${\bf R}^{n+1}$ of an $(n+1)$-component $n$-manifold-link $M_1 \cup \dots \cup M_{n+1}$ in ${\bf R}^{n+2}$; namely, a diagram has generic $(n+1)$-tuple points and we count the number of times $M_1$ is over $M_2$ is over ... is over $M_{n+1}$ with signs. It is difficult to show that this value is an invariant of the $n$-manifold-link in ${\bf R}^{n+2}$ directly, since we do not know Reidemeister moves for higher dimensions $(n \geq 3)$. However, the proof of Lemma 3.5 goes through to show that ${\mbox{\rm Glk}}$ is the same as this count (up to sign). Thus we have that this number (generalization of ${\mbox{\rm Tlk}}_1$) is an invariant of an $(n+1)$-component $n$-manifold-link.

[**Remark 6.2**]{} In classical link theory, the linking number determines the link homology classes completely. However, the triple linking of surface-links is [*not*]{} a complete invariant of the surface-link homology; there exists a pair of surface-links with the same triple linking invariants which are not link homologous. A classification of surface-link homology classes is discussed in a forthcoming paper.

[**Acknowledgments**]{} The authors would like to thank Akio Kawauchi for helpful suggestions. JSC is being supported by NSF grant DMS-9988107. SK and SS 