Translator Disclaimer
2018 On the asymptotic expansion of the quantum $\mathrm{SU}(2)$ invariant at $q = \exp(4\pi\sqrt{-1}/N)$ for closed hyperbolic $3$–manifolds obtained by integral surgery along the figure-eight knot
Tomotada Ohtsuki
Algebr. Geom. Topol. 18(7): 4187-4274 (2018). DOI: 10.2140/agt.2018.18.4187

Abstract

It is known that the quantum SU ( 2 ) invariant of a closed 3 –manifold at q = exp ( 2 π 1 N ) is of polynomial order as N . Recently, Chen and Yang conjectured that the quantum SU ( 2 ) invariant of a closed hyperbolic 3 –manifold at q = exp ( 4 π 1 N ) is of order exp ( N ς ( M ) ) , where ς ( M ) is a normalized complex volume of M . We can regard this conjecture as a kind of “volume conjecture”, which is an important topic from the viewpoint that it relates quantum topology and hyperbolic geometry.

In this paper, we give a concrete presentation of the asymptotic expansion of the quantum SU ( 2 ) invariant at q = exp ( 4 π 1 N ) for closed hyperbolic 3 –manifolds obtained from the 3 –sphere by integral surgery along the figure-eight knot. In particular, the leading term of the expansion is exp ( N ς ( M ) ) , which gives a proof of the Chen–Yang conjecture for such 3 –manifolds. Further, the semiclassical part of the expansion is a constant multiple of the square root of the Reidemeister torsion for such 3 –manifolds. We expect that the higher-order coefficients of the expansion would be “new” invariants, which are related to “quantization” of the hyperbolic structure of a closed hyperbolic 3 –manifold.

Citation

Download Citation

Tomotada Ohtsuki. "On the asymptotic expansion of the quantum $\mathrm{SU}(2)$ invariant at $q = \exp(4\pi\sqrt{-1}/N)$ for closed hyperbolic $3$–manifolds obtained by integral surgery along the figure-eight knot." Algebr. Geom. Topol. 18 (7) 4187 - 4274, 2018. https://doi.org/10.2140/agt.2018.18.4187

Information

Received: 15 January 2018; Revised: 29 June 2018; Accepted: 4 August 2018; Published: 2018
First available in Project Euclid: 18 December 2018

zbMATH: 07006390
MathSciNet: MR3892244
Digital Object Identifier: 10.2140/agt.2018.18.4187

Subjects:
Primary: 57M27
Secondary: 57M50

Rights: Copyright © 2018 Mathematical Sciences Publishers

JOURNAL ARTICLE
88 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.18 • No. 7 • 2018
MSP
Back to Top