respect to a projection $p:{\bf R}^4\rightarrow{\bf R}^3$. Let $\Gamma(F^*)$ denote the double point set of $F^*$; $$\{ p(x)\ |\  x \in F, \ p(x)=p(y) \ \mbox{for some} \
  y \in F, x \neq y \},$$ which consists of immersed curves, called [*double curves*]{}. A double curve $C^*$ is an immersed circle or an immersed arc in ${\bf R}^3$. If $C^*$ is an immersed circle, then $(p|_F)^{-1}(C^*)=C\cup C'$ for some pair of immersed circles $C$ and $C'$ in $F$. If $C^*$ is an immersed arc, then its endpoints are branch points of $F^*$ and $(p|_F)^{-1}(C^*)=C\cup C'$ for some pair of immersed arcs $C$ and $C'$ in $F$ with $\partial C= \partial C'$. The curves $C$ and $C'$ are called [*decker curves*]{} over $C^*$: one of them is in higher position than the other with respect to the projection direction, which is called an [*upper decker curve*]{} and the other is called an [*lower decker curve*]{}. We notice that the preimage of a triple point consists of three points of $F$ which are intersections of decker curves. See [@CS:book] for details.

Double curves and decker curves are oriented as follows: Let $x$ be a point of $F$ whose image $x^*=p(x)$ is not 