  We obtain a formula for the Turaev-Viro invariants of a link complement in
terms of values of the colored Jones polynomial of the link. As an application
we give the first examples for which the volume conjecture of Chen and the
third named author\,\cite{Chen-Yang} is verified. Namely, we show that the
asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the
Borromean rings complement determine the corresponding hyperbolic volumes. Our
calculations also exhibit new phenomena of asymptotic behavior of values of the
colored Jones polynomials that seem not to be predicted by neither the
Kashaev-Murakami-Murakami volume conjecture and various of its generalizations
nor by Zagier's quantum modularity conjecture. We conjecture that the
asymptotics of the Turaev-Viro invariants of any link complement determine the
simplicial volume of the link, and verify it for all knots with zero simplicial
volume. Finally we observe that our simplicial volume conjecture is stable
under connect sum and split unions of links.
