## Algebraic & Geometric Topology

### 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

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4187-4274.

Dates
Revised: 29 June 2018
Accepted: 4 August 2018
First available in Project Euclid: 18 December 2018

https://projecteuclid.org/euclid.agt/1545102070

Digital Object Identifier
doi:10.2140/agt.2018.18.4187

Mathematical Reviews number (MathSciNet)
MR3892244

Zentralblatt MATH identifier
07006390

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds
Secondary: 57M50: Geometric structures on low-dimensional manifolds

#### Citation

Ohtsuki, Tomotada. 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 (2018), no. 7, 4187--4274. doi:10.2140/agt.2018.18.4187. https://projecteuclid.org/euclid.agt/1545102070

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