second edges as specified by the stub.

Figure \[2jnetwork\] also gives the numerical values of the components of $K$, and in the final diagram, the network for $K$ itself, with two kets inserted. The network for $K$ in isolation is illustrated in Fig. \[2jKdef\], regarded as a bra vector on $\CS_{
j}\otimes\CS'_j$, that is, as an element of $\CS^*_j\otimes\CS^{\prime *}_j$.

The stub in Fig. \[2jnetwork\] can be regarded as a vestigial edge of a $3j$-symbol or $W$-intertwiner with the value $j=0$, although one must beware of the normalization convention. This is illustrated in Fig. \[vestigialstub\], which is equivalent to (\[2jdef\]). The value does not depend on the direction of the arrow on the zero edge (see below for rules for reversing arrows).

The components of $K:\CS_j\otimes\CS'_j\to\Complexes$, seen in Fig. \[2jnetwork\], acquire a phase of $(-1)^{2j}$ if $m$ and $m'$ are swapped. This is equivalent to the statement $$K(\ket{\phi},\ket{\psi}) = (-1)^{2j} K(\ket{\psi},\ket{\phi}),
    \label{Kswapphase}$$ for all $\ket{\psi}$, $\ket{\phi}$, which is illustrated in spin network language in Fig. \[Ksymm\]. The final diagram differs from the preceding simply by a $180^\circ$ rotation, so the value is the same. But this leads to the rule, that a spin network acquires a phase of $(-1)^{2j}$ when 