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The $3j$-symbol is a number that can be regarded as the components of an intertwiner $W:\CS_{j_1}\otimes\CS_{j_2}\otimes\CS_{j_3
}\to\Complexes$ with respect to the standard basis $\ket{j_1m_1
}\otimes\ket{j_2m_2}\otimes\ket{j_3m_3}$, as indicated in Fig. \[3jnetwork\]. In this paper $\CS_j$ denotes a carrier space for unitary irrep $j$ of $SU(2)$, so that $\dim\CS_j = 2j+1$. We call $W$ the “Wigner” intertwiner. We will gradually explain the features of Fig. \[3jnetwork\] as we proceed.

The standard notation for the $3j$-symbol is on the left of Fig. \[3jnetwork\], while the central diagram is the standard Yutsis spin network for the $3j$-symbol, with small arrows presented as in the Yutsis notation. The indices $(m_1,m_2,m_3)$ in the central diagram are covariant, that is, they transform under rotations as the components of a dual vector (in contrast to an ordinary vector). In a Hilbert space we regard ordinary wave functions or ket vectors as “vectors,” while bra vectors are regarded as “dual vectors.” Thus, contravariant indices are those that transform as the components of a vector. In the Yutsis notation the arrows indicate the transformation properties of the corresponding $m$ index, and there are rules for “raising and lowering” indices, that is, reversing the direction of the arrow. The rules 