## Algebraic & Geometric Topology

### Analytic families of quantum hyperbolic invariants

#### Abstract

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 $ℋNhf,hc,kc$ 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 $PSL(2, ℂ)$–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 $ℋNhf,hc,kc$ depend on the weights as $N →∞$, and recover the volume for some specific choices of the weights.

#### Article information

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

Dates
Revised: 25 September 2014
Accepted: 25 October 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.agt/1510840997

Digital Object Identifier
doi:10.2140/agt.2015.15.1983

Mathematical Reviews number (MathSciNet)
MR3402335

Zentralblatt MATH identifier
1335.57017

#### Citation

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. https://projecteuclid.org/euclid.agt/1510840997

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