Generating series and asymptotics of classical spin networks

We study classical spin networks with group SU(2). In the first part, using gaussian integrals, we compute their generating series in the case where the networks are equipped with holonomies; this generalizes Westbury's formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.Comment: 33 pages, 3 figures; in version 2 added one reference and a comment on the hypotheses of Theorem 1.

On SL(2, C) quantum 6j-symbol and its relation to the hyperbolic volume

We generalize the colored Alexander invariant of knots to an invariant of graphs, and we construct a face model for this invariant by using the corresponding 6j-symbol, which comes from the non-integral representations of the quantum group U_q(sl_2). We call it the SL(2, C) quantum 6j-symbol, and show its relation to the hyperbolic volume of a truncated tetrahedron.Comment: 30 pages, typos, argument for the R-matrix is modified, Sections 2 and 3 are exchange

Motion among random obstacles on a hyperbolic space

We consider the motion of a particle along the geodesic lines of the Poincar\`e half-plane. The particle is specularly reflected when it hits randomly-distributed obstacles that are assumed to be motionless. This is the hyperbolic version of the well-known Lorentz Process studied by Gallavotti in the Euclidean context. We analyse the limit in which the density of the obstacles increases to infinity and the size of each obstacle vanishes: under a suitable scaling, we prove that our process converges to a Markovian process, namely a random flight on the hyperbolic manifold.Comment: 19 pages, 4 figure

6j-symbols, hyperbolic structures and the Volume Conjecture

We compute the asymptotical growth rate of a large family of $U_q(sl_2)$ $6j$-symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We address a question which is strictly related with S.Gukov's generalized volume conjecture and deals with the case of hyperbolic links in connected sums of $S^2\times S^1$. We answer this question for the infinite family of fundamental shadow links.Comment: 17 pages, 3 figures. Published on Geometry & Topology 11 (2007

Some remarks on the unrolled quantum group of sl(2)

In this paper we consider the representation theory of a non-standard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl(2). The results of this paper are used extensively in [arXiv:1404.7289] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT).Comment: 25 pages, v3: several mistakes corrected in the formulas of modified trace