Turaev–Viro invariants, colored Jones polynomials, and volume

Abstract

We obtain a formula for the Turaev–Viro invariants of a link complement in terms of values of the colored Jones polynomials of the link. As an application, we give the first examples of 3-manifolds where the “large $r$” asymptotics of the Turaev–Viro invariants determine the hyperbolic volume. We verify the volume conjecture of Chen and the third named author [7] for the figure-eight knot and the Borromean rings. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem to be predicted neither by the Kashaev-Murakami-Murakami volume conjecture and 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 this conjecture for all knots with zero simplicial volume. Finally, we observe that our simplicial volume conjecture is compatible with connected summations and split unions of links.