consistency we must show that the two results are the same. This is done in Fig. \[Kconsist\], which uses the identity of Fig. \[2stubs\].

By inserting resolutions of the identity into the diagram in Fig. \[K1invreq\] it is easy to work out the components of $K^{-1}$. These are displayed in Fig. \[Kinvcomponents\]. Notice that they have the same numerical values as the components of $K$ (see Fig. \[2jnetwork\]).

By using $K^{-1}$ we can convert a bra chevron into a ket chevron on any spin network, not only on bras and kets themselves. This is illustrated in Fig. \[raiseindex\], which may be compared to Fig. \[lowerindex\].

Finally, by using Figs. \[lowerindex\], \[raiseindex\] and \[2stubs\], it may be shown that when we reverse the arrow on an edge of a spin network, we incur a phase of $(-1)^{2j}$. This is done in Fig. \[reversearrow\].

Raising and lowering indices {#raisingandlowering}
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When we convert a ket to a bra by the action of $K_1$, then the bra has components with respect to the standard basis $\ket{jm}$ that are simple functions of the components of the original ket with respect to the standard basis $\bra{jm}$. Mapping the one set of components to the other 