The volume conjecture for hyperbolic polyhedra

Let $P$ be a hyperbolic polyhedron with dihedral angles $\alpha_1,\dots,\alpha_m$ at edges $e_1,\dots,e_m$, and let $\Gamma$ be its $1$-skeleton. Let $col_r$ be a sequence of $r$-admissible colorings of these edges such that

$$2\pi\lim_{r\rightarrow+\infty}\frac{col_r(e_i)}{r}=\pi-\alpha_i.$$

Here $Y_r(S^3,\Gamma,col_r,e^{2\pi i/r})$ denotes the Yokota invariant of the graph with these edge colorings. The volume conjecture for polyhedra. Then

$$\lim_{r\rightarrow+\infty}\frac{\pi}{r}\log\left\lvert Y_r(S^3,\Gamma,col_r,e^{2\pi i/r})\right\rvert=\operatorname{Vol}(P).$$

This conjecture connects asymptotic quantum invariants of planar graphs with hyperbolic volume. It generalizes earlier volume conjectures for quantum $6j$-symbols, trivalent graphs, and simple polyhedra; the paper uses it to derive the weak Stoker conjecture.