do not, however, make use of the metric as in ordinary tensor analysis. Our definition of the arrow (explained below) is different from that of Yutsis, but designed so that the two notations agree as much as possible. In particular, our notation for the $3j$-symbol is the same as the Yutsis diagram in Fig. \[3jnetwork\].

A trivalent node of a spin network such as those illustrated in Fig. \[3jnetwork\] is assumed to have a positive or counterclockwise orientation, unless otherwise indicated (thus we dispense with the $+$ sign used by Yutsis).

Bras, kets and scalar products {#brasketssps}
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The diagram on the right of Fig. \[3jnetwork\] makes use of the standard basis vectors $\ket{j_im_i}$ in $\CS_{j_i}$, $i=1,2,3$. The spin networks for these basis vectors and their duals are shown in Fig. \[basisvectors\]. The large, broadly open arrow is a “chevron” (Stedman 1990). When pointing outward (inward), the chevron indicates a ket (bra) vector. Also shown in Fig. \[basisvectors\] is the spin network for an arbitrary vector $\ket{\psi}$ in $\CS_j$, and the dual bra vector $\bra{\psi}$ obtained by Hermitian conjugation. In the Dirac notation it is customary to label bras by the same symbol as kets, it being understood that the 