Yang–Wong volume 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. For odd integers $r\geq 3$, let $\mathbf m=(m_1,\dots,m_n)$ and let $RT_r(M,L,\mathbf m)$ be the $r$-th relative Reshetikhin–Turaev invariant of $M$ with $L$ colored by $\mathbf m$, evaluated at $q=e^{\frac{2\pi\sqrt{-1}}{r}}$. For a sequence $\mathbf m^{(r)}=(m_1^{(r)},\dots,m_n^{(r)})$, define

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

If $M_{L_\theta}$ is the hyperbolic cone manifold consisting of $M$ equipped with a hyperbolic cone metric whose singular locus is $L$ and whose cone angles are $\theta$, then Yang–Wong's volume conjecture.

$$\lim_{r\to\infty}\frac{4\pi}{r}\log RT_r(M,L,\mathbf m^{(r)})=\operatorname{Vol}(M_{L_\theta})+\sqrt{-1}\operatorname{CS}(M_{L_\theta}),$$

where $r$ ranges over all positive odd integers. This conjecture predicts that the asymptotics of the relative invariants recover the hyperbolic volume and Chern–Simons invariant of the associated cone manifold. The paper verifies the leading-order term for pairs whose link complements are fundamental shadow link complements after suitable rational Dehn fillings, under additional assumptions on surgery denominators and sufficiently small cone angles.