genera of the components of $F$. By Theorem 1.3, we see that

\(1) if $a\ne 0$ and $b=0$, then $g(K_i)\geq 1$ $(i=1,2)$, and

\(2) if $a\ne 0$, $b\ne 0$ and $a+b\ne 0$, then $g(K_i)\geq 1$ $(i=1,2,3)$,

where $g(K_i)$ denotes the genus of $K_i$.

[(i)]{} For any integer $a\ne 0$, there exists a surface-link $F=K_1\cup K_2\cup K_3$ whose triple linking numbers satisfy the above equations $(*)$ with $b=0$ and $g(K_i)=1$ $(i=1,2)$ and $g(K_3)=0$.

[(ii)]{} For any pair of integers $a$ and $b$ with $a\ne 0$, $b\ne 0$ and $a+b\ne 0$, there exists a surface-link $F=K_1\cup K_2\cup K_3$ whose triple linking numbers satisfy the above equations $(*)$ and $g(K_i)=1$ $(i=1,2,3)$.

This paper is organized as follows: in Section 2, we interprete ${\mbox{\rm Tlk}}_1$ in terms of the decker curves of a surface diagram. In Section 3 we give precise definitions of triple linking numbers ${\mbox{\rm Tlk}}_i$ for $i=2,\dots,5$ (in terms of homology) and prove Theorem 1.1. Section 4 is devoted to proving Theorems 1.2–1.5. Proposition 1.6 is proved in Section 5.

Throughout this paper, all the homology and cohomology groups have the [**Z**]{}-coefficient.

Decker Curves and Triple Linking
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Let $F$ be a surface-link and $F^*$ a surface diagram of $F$ with 