#### Volume 17, issue 1 (2013)

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Asymptotics of classical spin networks

### Appendix: Don Zagier

Geometry & Topology 17 (2013) 1–37
##### Abstract

A spin network is a cubic ribbon graph labeled by representations of $SU\left(2\right)$. Spin networks are important in various areas of Mathematics ($3$–dimensional Quantum Topology), Physics (Angular Momentum, Classical and Quantum Gravity) and Chemistry (Atomic Spectroscopy). The evaluation of a spin network 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) rigorous effective computations of our results for some $6j$–symbols using the Wilf–Zeilberger theory and (d) a complete analysis of the regular Cube $12j$ spin network (including a nonrigorous guess of its Stokes constants), in the appendix.

##### Keywords
Spin networks, ribbon graphs, $6j$–symbols, Racah coefficients, angular momentum, asymptotics, G-functions, Kauffman bracket, Jones polynomial, Wilf-Zeilberger method, Borel transform, enumerative combinatorics, recoupling, Nilsson
Primary: 57N10
Secondary: 57M25