Abstract. The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas. However, we show numerically that the dominant contribution comes from degenerate 4-simplices. As a consequence, one can compute the asymptotics of the Riemannian 10j symbols by evaluating a 'degenerate spin network', where the rotation group SO (4) is replaced by the Euclidean group of isometries of R 3 . We conjecture formulas for the asymptotics of a large class of Riemannian and Lorentzian spin networks in terms of these degenerate spin networks, and check these formulas in some special cases. Among other things, this conjecture implies that the Lorentzian 10j symbols are asymptotic to 1/16 times the Riemannian ones.

Abstract. The 10j symbol is a spin network that appears in the partition function for the Barrett-Crane model of Riemannian quantum gravity. Elementary methods of calculating the 10j symbol require O(j 9 ) or more operations and O(j 2 ) or more space, where j is the average spin. We present an algorithm that computes the 10j symbol using O(j 5 ) operations and O(j 2 ) space, and a variant that uses O(j 6 ) operations and a constant amount of space. An implementation has been made available on the web.