by its use of what we call the “$4j$-model” for the $6j$-symbol, in contrast to the “$12j$-model” used by Roberts (1999). The $4j$-model is less symmetrical than the $12$-model, but it is closer to the manner in which the $6j$-symbol is commonly used in recoupling theory. In addition, in loop quantum gravity (Rovelli 2004) angular momenta represent area vectors, which in the case of four-valent nodes correspond by Minkowski’s (1897) theorem to a tetrahedron. In this context the $4j$-model is closer to the applications than the $12j$-model, indeed, it played an important role in the work of Bianchi and Haggard (2011).

In this paper we refer to Aquilanti (2007) as I, for example writing eqn. (I.13) for an equation from that paper. We note two errata in I, namely, $\sigma(x)$ in (I.89) should read $\sigma(u)$, and $j_3$ and $j_4$ should be swapped in the $6j$-symbol in (I.112).

Spin network notation {#spinnetworknotation}
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We begin by explaining our notation for spin networks, which is based on that of Yutsis (1962) with modifications due to Stedman (1990). At the end of this section we compare our conventions for spin networks with others in the Yutsis tradition.

The $3j$-symbol and Wigner intertwiner {#3jwigner}
