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

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.