{\bf R}^3$ with $p(w,x,y,z) = (x,y,z)$. The preimage of a particular point $((1,0,0,0), (1,0,0,0))$ by $L$ consists of triples $(x_1,x_2,x_3) \in F_1 \times F_2 \times
F_3$ such that $p(f(x_1))= p(f(x_2)) = p(f(x_3))$ and $f(x_1)$ is the upper, $f(x_2)$ is the middle, $f(x_3)$ is the lower lift of the triple point $p(f(x_1))$. For each such triple $(x_1, x_2, x_3)$, let $D_T^2$, $D_M^2$, $D_B^2$ be regular neighborhoods of them in $F_1 \cup F_2 \cup F_3$, and let $\varepsilon \in \{+1, -1\}$ be the sign of the triple point $p(f(x_1))$. Let $(x_T,y_T)$, $(x_M, -\varepsilon z_M)$ and $(y_B,z_B)$ be coordinate systems of $D_T^2$, $D_M^2$ and $D_B^2$ around $x_1, x_2$ and $x_3$, respectively. Modifying $f$ up to ambient isotopy, we may assume that the restriction of $f$ to $D_T^2 \cup D_M^2 \cup D_B^2$ is given by defined by $$\begin{aligned}
(x_T,y_T) &
\mapsto & (0, x_0, y_0, z_0)+ (3,x_T,y_T,0) \\
(x_M,- \varepsilon z_M) &
\mapsto & (0, x_0, y_0, z_0)+ (2,x_M, 0, z_M) \\
(y_B,z_B) &
\mapsto & (0, x_0, y_0, z_0)+ (1,y_B, z_B) \end{aligned}$$ where $(x_0, y_0, z_0) \in {\bf R}^3$ is the triple point $p(f(x_1))$. In this situation, the restriction $$L': D_T^2 \times D_M^2 \times D_B^2 \rightarrow S^3 \times
S^3$$ is given by the 