On the asymptotic expansion of the quantum SU(2) invariant at q = exp(4π∕N) for closed hyperbolic 3–manifolds obtained by integral surgery along the figure-eight knot

Ohtsuki, Tomotada

doi:10.2140/agt.2018.18.4187

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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

DOI: 10.2140/agt.2018.18.4187

Abstract

It is known that the quantum $SU (2)$ invariant of a closed $3$ –manifold at $q = exp (2 π \sqrt{- 1} ∕ N)$ is of polynomial order as $N \to \infty$ . Recently, Chen and Yang conjectured that the quantum $SU (2)$ invariant of a closed hyperbolic $3$ –manifold at $q = exp (4 π \sqrt{- 1} ∕ N)$ is of order $exp (N \cdot ς (M))$ , 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 $q = exp (4 π \sqrt{- 1} ∕ N)$ 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 $exp (N \cdot ς (M))$ , 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/

Volume 18, issue 7 (2018)