two are related by Hermitian conjugation. This convention is so deeply ingrained that we dare not change it. But in spin networks there are two different ways of converting kets into bras and vice versa, and this presents some notational challenges. One can see in Fig. \[basisvectors\] that Hermitian conjugation applied to bras and kets is notationally the changing of bra chevrons to ket chevrons and vice versa, and the starring of identifying symbols, with a double star being removed. Full rules for Hermitian conjugation of any spin network are given in Sec. \[Hermitianconjugation\].

The lines of a spin network will be referred to as “edges,” including cases like those shown in Fig. \[basisvectors\].

An edge of a spin network ending in an unstarred $m$ index represents a contraction with the basis ket $\ket{jm}$, so the $m$ index transforms under rotations as a covariant index. The explicit insertion of basis kets may be seen in Fig. \[3jnetwork\]. An edge ending in a starred $m$ index represents the insertion of a basis bra $\bra{jm}$, so starred indices are contravariant. Thus, the star on an $m$ index indicates its transformation property, and invariant contractions can only take place between a pair of 