Résumé
We construct
quantum hyperbolic invariants (QHI) for triples (
W,
L,
ρ), where W is a compact closed oriented 3-manifold,
ρ is a flat principal bundle over W with structural group
PSL(2,
C)
, and L is a non-empty link in W. These invariants are based on the Faddeev–Kashaev's
quantum dilogarithms, interpreted as matrix-valued functions of suitably decorated hyperbolic ideal tetrahedra. They are explicitly computed as state sums over the decorated hyperbolic ideal tetrahedra of the
idealization of any fixed
D
-
triangulation; the
D
-triangulations are simplicial 1-cocycle descriptions of (
W,
ρ) 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.