the state-sum invariants of surface-links. In fact, the state-sum invariants in the classical link and surface-link case generalize linking number and Fox’s coloring number.

In the current paper, we give several alternative definitions of the triple linking number and some properties. The reader will find that this invariant is a quite natural generalization of the notion of classical linking number in contrast to a statement in Rolfsen [@Rolfsen] page 136: “There is, however no analogous notion of linking number to help us with codimension two link theory, for example, in higher dimensions”. We note, however, that Rolfsen himself with Massey [@MassayRolf] and with Fenn [@FennRolf] generalized classical linking numbers to higher dimensions using degrees of maps.

Let $F=K_1\cup K_2\cup K_3$ be a $3$-component surface-link in ${\bf R}^4$. It is known (see [@CS:book] for example) that a projection of $F$ into ${\bf R}^3$ can be assumed to have transverse double curves and isolated branch/triple points. At a triple point, three sheets intersect that have distinct relative heights with respect to the projection direction, and we call them top, middle, and bottom sheets, accordingly. If the orientation normals to the top, middle, bottom sheets at a triple point $\tau$ matches with this 