a branch point. There is a regular neighborhood $N$ of $x$ in $F$ such that $p|_N$ is an embedding. An [*orientation normal $\vec{n}$ to $N^*=p(N)$* ]{} in ${\bf R}^3$ at $x^*$ is specified in such a way that $(\vec{v}_1, \vec{v}_2, \vec{n})$ matches the orientation of ${\bf R}^3$, where the pair of tangents $(\vec{v}_1, \vec{v}_2)$ defines the orientation of $N^*$ that is induced from the orientation of $N \subset F$. If $y$ is a double point on a double curve $C^*$, then $C^*$ is locally an intersection of $N_1^*$ and $N_2^*$, where $N_1^*$ is upper and $N_2^*$ is lower. We assign a tangent vector $\vec{v}$ of $C^*$ at $y$ such that $(\vec{n}_1,
\vec{n}_2, \vec{v})$ matches the orientation of ${\bf R}^3$. This defines an orientation of $C^*$, cf. [@CJKLS; @CS:book]. We give an orientation to the lower decker curve over $C^*$ such that it inherits the orientation from $C^*$, and give the opposite orientation to the upper decker curve. Note that, if $C^*$ is an arc, then the orientations of $C$ and $C'$ are compatible (i.e., the union $C \cup C'$ forms an oriented immersed circle in $F$).

Let $F=K_1 \cup K_2 \cup K_3$ be a 3-component surface-link. A double curve $C^*$ 