of $\gamma$ in ${\bf R}^4$ and let $T_1 \cup T_2 \cup T_3$ be the image of $\ell \times S^1 =
k_1 \times S^1 \cup k_2 \times S^1 \cup k_3 \times S^1$ in ${\bf R}^4$. Let $F = K_1 \cup K_2 \cup K_3$ be a surface-link obtained from $(T_1 \cup T_2 \cup T_3) \cup (\partial B_1 \cup
\partial B_2 \cup \partial B_3)$ by piping such that $F$ has a projection as in Figure 3. Then $F$ is the desired link.  

Remarks
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[**Remark 6.1**]{} The definition of ${\mbox{\rm Tlk}}_6$ can be seen as a direct analogue of (6) given in [@Rolfsen] page 133. See also [@FennRolf; @MassayRolf]. This generalizes the triple linking to all link maps, instead of embeddings. Moreover, it easily generalized to all dimensions. Let $M_i$ denote a closed connected $n$-manifold for $i=1,\ldots , n+1$. Let an embedding $f: \cup_{i=1}^{n+1} M_i
\rightarrow  {\bf R}^{n+2}$ be given. Define $L: \prod_{i=1}^{n+1} M_i \rightarrow
\prod_{j=1}^n S^{n+1}_j$ as follows. Let $x_i \in M_i$; for $i=1, \ldots, n$, let $\Delta_i = f(x_i)-f(x_{i+1})/ || f(x_i)-f(x_{i+1}) ||$. Then $$L(x_1,\ldots x_{n+1}) = ( \Delta_1, \Delta_2, \ldots , \Delta_n).$$ The general $(n+1)$-fold linking number, ${\mbox{\rm Glk}}$, is defined by $${\mbox{\rm Glk}}(f(M_1), \dots, f(M_{n+1}))= \deg(L).$$ We can generalize the 