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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

Tomotada Ohtsuki

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

It is known that the quantum SU(2) invariant of a closed 3–manifold at q = exp(2π1N) 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(4π1N) 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(4π1N) 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.

Keywords
knot, $3$–manifold, quantum invariant, volume conjecture
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M50
References
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/