Algebraic & Geometric Topology

Analytic families of quantum hyperbolic invariants

Stéphane Baseilhac and Riccardo Benedetti

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We organize the quantum hyperbolic invariants (QHI) of 3–manifolds into sequences of rational functions indexed by the odd integers N 3 and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic 3–manifolds M we generalize the QHI and get rational functions Nhf,hc,kc depending on a finite set of cohomological data (hf,hc,kc) called weights. These functions are regular on a determined Abelian covering of degree N2 of a Zariski open subset, canonically associated to M, of the geometric component of the variety of augmented PSL(2, )–characters of M. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions Nhf,hc,kc depend on the weights as N , and recover the volume for some specific choices of the weights.

Article information

Algebr. Geom. Topol., Volume 15, Number 4 (2015), 1983-2063.

Received: 1 March 2014
Revised: 25 September 2014
Accepted: 25 October 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M27: Invariants of knots and 3-manifolds 57Q15: Triangulating manifolds
Secondary: 57R56: Topological quantum field theories

quantum invariants 3–manifolds character varieties Chern–Simons theory volume conjecture


Baseilhac, Stéphane; Benedetti, Riccardo. Analytic families of quantum hyperbolic invariants. Algebr. Geom. Topol. 15 (2015), no. 4, 1983--2063. doi:10.2140/agt.2015.15.1983.

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