the set of triple points of type $(1,2,3)$ and double points of $D_{12}^{\ell}$ and $D_{23}^u$. If the sign of $\tau$ is positive (or negative, resp.) then the corresponding intersection of $D_{12}^{\ell}$ and $D_{23}^u$ is negative (resp. positive), see Figure 1. Thus we have the result.  ${\hbox{\rule{6pt}{6pt}}}$

Since $D_{12}^{\ell}$ is the union of circles in ${\bf R}^4$ disjoint from $K_3$, the linking number ${\mbox{\rm Link}}(D_{12}^{\ell}, K_3)$ is defined.

${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)={\mbox{\rm Link}}(D_{12}^{\ell}, K_3).$

[*Proof.*]{} Without loss of generality, we may assume that the projection $p$ is given by $p(w,x,y,z) = (x,y,z)$. For a real number $\lambda$ we denote by $t_\lambda: {\bf R}^4 \to {\bf R}^4$ the translation with $t_\lambda(w,x,y,z)= (w+\lambda, x,y,z)$. Let $M'_3$ be a 3-chain in ${\bf R}^4$ with $\partial M'_3 = K_3$. Take a sufficiently large number $R$ and consider a $3$-chain $$M_3 = \cup_{\lambda \in [0, R]} t_\lambda(K_3)
+ t_{R}(M'_3)$$ so that $\partial M_3= K_3$ and $D_{12}^{\ell} \cap M_3 = D_{12}^{\ell}
\cap (\cup_{\lambda \in [0, R]}
t_\lambda(K_3))$. The projection $p$ induces a one-to-one correspondence between the geometric intersection $D_{12}^{\ell} \cap (\cup_{\lambda \in [0, R]}
t_\lambda(K_3))$ and the subset of $D_{12}^{\ell *} \cap K_3^* = p(D_{12}^{\ell}) \cap p(K_3)$ consisting of points where $D_{12}^{l*}$ is higher than $K_3^*$ (in the over-under 