Stavros Garoufalidis and Roland van der Veen
Abstract: A spin network consists of a cubic ribbon graph (i.e., an abstract trivalent graph with a cyclic ordering of the vertices around each edge) and an admissible coloring of its edges by natural numbers. The evaluation of a spin network, with our normalization, is an integer number. The main results of our paper are: (a) an existence theorem for the asymptotics of evaluations of arbitrary spin networks (using the theory of $G$-functions), (b) a rationality property of the generating series of all evaluations with a fixed underlying graph (using the combinatorics of the chromatic evaluation of a spin network), (c) a complete study of the asymptotics of the so-called $6j$-symbols in all cases: Euclidean, Plane or Minkowskian (using the theory of Borel transform), (d) an assignement of two numerical invariants (a spectral radius and a number field) to spin networks and cubic graphs and (e) an explicit illustration of our results for some Euclidean, Plane and Minkowskian $6j$-symbols (using the constructive Wilf-Zeilberger method).
Key words: Spin networks, ribbon graphs, $6j$-symbols, Racah coefficients, angular momentum, asymptotics, $G$-functions, Kauffman bracket, Jones polynomial, Wilf-Zeilberger method, Borel transform, enumerative combinatorics.
Notes: 28 pages, 28 figures.