#### Volume 18, issue 7 (2018)

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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

Algebraic & Geometric Topology 18 (2018) 4187–4274
##### Abstract

It is known that the quantum $SU\left(2\right)$ invariant of a closed $3$–manifold at $q=exp\left(2\pi \sqrt{-1}∕N\right)$ is of polynomial order as $N\to \infty$. Recently, Chen and Yang conjectured that the quantum $SU\left(2\right)$ invariant of a closed hyperbolic $3$–manifold at $q=exp\left(4\pi \sqrt{-1}∕N\right)$ is of order $exp\left(N\cdot \varsigma \left(M\right)\right)$, where $\varsigma \left(M\right)$ is a normalized complex volume of $M\phantom{\rule{0.3em}{0ex}}$. 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 $SU\left(2\right)$ invariant at $q=exp\left(4\pi \sqrt{-1}∕N\right)$ 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 $exp\left(N\cdot \varsigma \left(M\right)\right)$, 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.

##### Keywords
knot, $3$–manifold, quantum invariant, volume conjecture
Primary: 57M27
Secondary: 57M50
##### Publication
Received: 15 January 2018
Revised: 29 June 2018
Accepted: 4 August 2018
Published: 11 December 2018
##### Authors
 Tomotada Ohtsuki Research Institute for Mathematical Sciences Kyoto University Sakyo-ku, Kyoto Japan http://www.kurims.kyoto-u.ac.jp/~tomotada/