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 $SU\left(2\right)$ 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 $M$. 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.