order the fixed orientation of ${\bf R}^3$, then the sign of $\tau$ is positive and $\varepsilon(\tau)=1$. Otherwise the sign is negative and $\varepsilon(\tau)=-1$. (See [@CJKLS; @CS:book].) It is also known that any closed oriented embedded surface $F$ in ${\bf R}^4$ bounds an oriented compact $3$-manifold $M$ embedded in ${\bf R}^4$, called a [*Seifert hypersurface*]{} of $F$, such that $\partial M=F$.

We give six methods for defining an integer (triple linking number) as follows.

-   Consider a surface diagram of $K_1\cup K_2\cup K_3$ in ${\bf R}^3$. A triple point is [*of type $(i,j,k)$*]{} if the top sheet comes from $K_i$, the middle comes from $K_j$, and the bottom comes from $K_k$. The sum of the signs of all the triple points of type $(1,2,3)$ is denoted by ${\mbox{\rm Tlk}}_1(K_1,K_2,K_3)$. This is the definition given in [@CJKLS].

-   Let $M_i$ be a Seifert hypersurface for $K_i$ $(i=1,3)$. Assume that $M_i\cap K_2$ is a $1$-manifold in $K_2$ and that $M_1\cap K_2$ and $M_3\cap K_2$ intersect transversely. Count the intersections between them algebraically and denote the sum by ${\mbox{\rm Tlk}}_2(K_1,K_2,K_3)$.

-   Consider a Seifert hypersurface $M_1$ for $K_1$. Assume that $M_1\cap K_2$ is a $1$-manifold, which is disjoint from $K_3$. The linking number ${\mbox{\rm 