in atomic, molecular and nuclear physics. These usually involve the recoupling of three angular momenta, that is, the $6j$-symbol contains the unitary matrix elements of the transformation connecting the two bases that arise when three angular momenta are added in two different ways. Such applications and the definition of the $6j$-symbol based on them are described by Edmonds (1960). More recently the $6j$- and other $3nj$-symbols have found applications in quantum computing (Marzuoli and Rasetti 2005) and in algorithms for molecular scattering calculations (De Fazio  2003, Anderson and Aquilanti 2006), which make use of their connection with discrete orthogonal polynomials (Aquilanti  1995, 2001a,b).

The $6j$-symbol is an example of a spin network, a graphical representation for contractions between tensors that occur in angular momentum theory. The graphical notation has been developed by Yutsis (1962), El Baz and Castel (1972), Lindgren and Morrison (1986), Varshalovich (1981), Stedman (1990), Danos and Fano (1998), Wormer and Paldus (2006), Balcar and Lovesey (2009) and others. The $6j$-symbol is the simplest, nontrivial, closed spin network (one that represents a rotational invariant). Spin networks are important in lattice QCD and in loop quantum gravity where they provide a gauge-invariant basis for the field. Applications in quantum 