In this paper, we show that the Turaev–Viro invariant
volume conjecture posed by Chen and Yang is preserved
under gluings of toroidal boundary components for a
family of 3-manifolds. In particular, we show that the
asymptotics of the Turaev–Viro invariants are additive
under certain gluings of elementary pieces arising from
a construction of hyperbolic cusped 3-manifolds due to
Agol. The gluings of the elementary pieces are known to
be additive with respect to the simplicial volume. This
allows us to construct families of manifolds which have
an arbitrary number of hyperbolic pieces and satisfy an
extended version of the Turaev–Viro invariant volume
conjecture.
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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.
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- Award ID(s):
- 1708249
- NSF-PAR ID:
- 10156667
- Date Published:
- Journal Name:
- Quantum topology
- Volume:
- 9
- Issue:
- 4
- ISSN:
- 1663-487X
- Page Range / eLocation ID:
- 775-813
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
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