It is known that the quantum
invariant of a closed
–manifold
at
is of polynomial
order as
.
Recently, Chen and Yang conjectured that the quantum
invariant of a closed
hyperbolic
–manifold
at
is of order
, where
is a normalized
complex volume of
.
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
invariant at
for closed hyperbolic
–manifolds obtained
from the
–sphere
by integral surgery along the figure-eight knot. In particular, the leading term of the
expansion is
,
which gives a proof of the Chen–Yang conjecture for such
–manifolds.
Further, the semiclassical part of the expansion is a constant
multiple of the square root of the Reidemeister torsion for such
–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
–manifold.
