gravity are described by Rovelli and Smolin (1995), Baez (1996), Carlip (1998), Barrett and Crane (1998), Regge and Williams (2000), Rovelli (2004) and Thiemann (2007), among others.

Alongside the Yutsis school of graphical notation and the Clebsch-Gordan school of algebraic manipulation there is a third approach to the evaluation of rotational ($SU(2)$) invariants. The third method, sometimes called chromatic evaluation, grew out of Penrose’s doctoral work on the graphical representation of tensors and is closely related to knot theory. We will not have further occasion to mention this school, see Penrose (1971) for its introduction, Rovelli (2004) for an overview and Kauffman and Lins (1994) for its full development.

The asymptotics of spin networks and especially the $6j$-symbol has played an important role in many areas. By “asymptotics” we refer to the asymptotic expansion for the spin network when all $j$’s are large, equivalent to a semiclassical approximation since large $j$ is equivalent to small $\hbar$. The asymptotic expression for the $6j$-symbol (the leading term in the asymptotic series) was first obtained by Ponzano and Regge (1968), or, more precisely, they obtained several formulas, valid inside and outside the classically allowed region and in the neighborhood of the caustics. In 