On the asymptotic expansion of various quantum invariants III: the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral surgery along the twist knot

This is the third article in a series devoted to the study of the asymptotic expansions of various quantum invariants related to the twist knots. In this paper, by using the saddle point method developed by Ohtsuki and Yokota, we obtain an asymptotic expansion formula for the Reshetikhin-Turaev invariants of closed hyperbolic 3-manifolds obtained by doing integral $q$-surgery along the twist knots ${\mathcal{K}}_p$ at the root of unity $e^{\frac{4\pi \sqrt{-1}}{r}}$ ($r$ is odd).