Abstract
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.
Citation
Tomotada Ohtsuki. "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." Algebr. Geom. Topol. 18 (7) 4187 - 4274, 2018. https://doi.org/10.2140/agt.2018.18.4187
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