Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

Details Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters Quantum Hyperbolic Invariants Of 3-Manifolds With PSL(2,C)-Characters

2003-06-19

arxiv.org/abs/math/0306280v1

Topology 43 (2004) 1373--1423

Mathematics

Geometric Topology

49 pages, 17 figures. Together with our paper `Classical And Quantum Dilogarithmic Invariants Of Flat PSL(2,C)-Bundles Over 3-

Scientific paper

We construct {\it quantum hyperbolic invariants} (QHI) for triples $(W,L,\rho)$, where $W$ is a compact closed oriented 3-manifold, $\rho$ is a flat principal bundle over $W$ with structural group $PSL(2,\mc)$, and $L$ is a non-empty link in $W$. These invariants are based on the Faddeev-Kashaev's {\it quantum dilogarithms}, interpreted as matrix valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitely computed as state sums over the decorated hyperbolic ideal tetrahedra of the {\it idealization} of any fixed {\it $\Dd$-triangulation}; the $\Dd$-triangulations are simplicial 1-cocycle descriptions of $(W,\rho)$ in which the link is realized as a Hamiltonian subcomplex. We also discuss how to set the Volume Conjecture for the coloured Jones invariants $J_N(L)$ of hyperbolic knots $L$ in $S^3$ in the framework of the general QHI theory.

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