Turaev-Viro invariants, colored Jones polynomial and volume.
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.
Authors:
Award ID(s):
Publication Date:
NSF-PAR ID:
10156667
Journal Name:
Quantum topology
Volume:
9
Issue:
4
Page Range or eLocation-ID:
775-813
ISSN:
1663-487X
1. We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants $TV_r$and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold $M$, with (possibly empty) toroidal boundary, $log|TVr(M)|$ is bounded above by a function linear in $r$ and whose slope is a positive universal constant times the Gromov norm of $M$. The proof combines TQFT techniques, geometric decomposition theory of 3-manifolds and analytical estimates of $6j$-symbols. We obtain topological criteria that can be used to check whether the growth is actually exponential; that is one has $log|TVr(M)|\geq B r$, for some $B>0$. We use these criteria to construct infinite families of hyperbolic 3-manifolds whose $SO(3)$- Turaev-Viro invariants grow exponentially. These constructions are essential for the results of article [3] where we make progress on a conjecture of Andersen, Masbaum and Ueno about the geometric properties of surface mapping class groups detected by the quantum representations. We also study the behavior of the Turaev-Viro invariants under cutting and gluing of 3-manifolds along tori. In particular, we show that, like the Gromov norm, the values of the invariants do not increase under Dehn filling and we give applications of this result on the question ofmore »