Yang–Wong asymptotic expansion conjecture for relative Reshetikhin–Turaev invariants

Let $M$ be a closed oriented $3$-manifold and let $L$ be a framed hyperbolic link in $M$ with $n$ components. Let $\mathbf m^{(r)}=(m_1^{(r)},\dots,m_n^{(r)})$ be colorings by elements of $\{0,\dots,r-2\}$ such that, for each $k$, either $m_k^{(r)}>r/2$ or $m_k^{(r)}<r/2$ for all sufficiently large $r$. Set $\mu_k=1$ in the former case and $\mu_k=-1$ in the latter, and define

$$\theta_k^{(r)}=\mu_k\left(\frac{4\pi m_k^{(r)}}{r}-2\pi\right),\qquad \theta^{(r)}=(\theta_1^{(r)},\dots,\theta_n^{(r)}).$$

Assume that, for all sufficiently large $r$, a hyperbolic cone metric on $M$ with singular locus $L$ and cone angles $\theta^{(r)}$ exists. Write $M^{(r)}$ for this cone manifold, let $\operatorname{Vol}(M^{(r)})$ and $\operatorname{CS}(M^{(r)})$ denote its volume and Chern–Simons invariant, let $\mathrm H^{(r)}(\gamma_k)$ be the logarithmic holonomy of the framed parallel copy $\gamma_k$ of the $k$-th core curve, and let $\mathbb T_{(M{\smallsetminus} L,\mathbf\Upsilon)}([\rho_{M^{(r)}}])$ be the Reidemeister torsion twisted by the adjoint holonomy representation with respect to meridians $\mathbf\Upsilon$. If $\{\theta^{(r)}\}$ converges, then Yang–Wong's asymptotic expansion conjecture. For odd positive integers $r$ and $q=e^{\frac{2\pi\sqrt{-1}}{r}}$,

$$RT_r(M,L,\mathbf m^{(r)})=C\frac{e^{\frac12\sum_{k=1}^n\mu_k\mathrm H^{(r)}(\gamma_k)}}{\sqrt{\pm\mathbb T_{(M{\smallsetminus} L,\mathbf\Upsilon)}([\rho_{M^{(r)}}])}}e^{\frac{r}{4\pi}(\operatorname{Vol}(M^{(r)})+\sqrt{-1}\operatorname{CS}(M^{(r)}))}\left(1+O\left(\frac1r\right)\right),$$

where $C$ has norm $1$ and is independent of the geometric structure on $M$. This refines the leading-order volume conjecture by predicting the torsion, holonomy, phase, and first asymptotic correction of the relative invariant. The paper verifies this leading asymptotic expansion in the fundamental-shadow-link and rational-Dehn-filling setting under the stated restrictions, while the conjecture itself is not established in the full generality stated.