Geometry & Topology

Asymptotics of classical spin networks

Stavros Garoufalidis and Roland van der Veen

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A spin network is a cubic ribbon graph labeled by representations of SU(2). 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.

Article information

Geom. Topol., Volume 17, Number 1 (2013), 1-37.

Received: 20 February 2009
Revised: 9 May 2012
Accepted: 12 July 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N10: Topology of general 3-manifolds [See also 57Mxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

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


Garoufalidis, Stavros; van der Veen, Roland. Asymptotics of classical spin networks. Geom. Topol. 17 (2013), no. 1, 1--37. doi:10.2140/gt.2013.17.1.

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