The volume conjecture for simple generalized hyperbolic polyhedra

Let PP) be a simple generalized hyperbolic polyhedron with dihedral angles θ1,,θm\theta_1,\dots,\theta_m at the edges e1,,eme_1,\dots,e_m, and 11-skeleton Γ\Gamma. Let colrcol_r be a sequence of rr-admissible colorings of the edges e1,,eme_1,\dots,e_m of Γ\Gamma such that

2πlimr+colr(ek)r=π±θk.2\pi\lim_{r\rightarrow+\infty}\frac{col_r(e_k)}{r}=\pi\pm\theta_k.

The volume conjecture for polyhedra. At q=e2π1rq=e^{\frac{2\pi\sqrt{-1}}{r}}, as rr runs over all the odd integers,

limr+2πrlnΓ,colrq=Vol(P).\lim_{r\rightarrow+\infty}\frac{2\pi}{r}\ln\big|\langle\Gamma,col_r\rangle_q\big|=\operatorname{Vol}(P).

This extends volume-conjecture formulations for simple hyperideal and compact polyhedra to simple generalized hyperbolic polyhedra, with regular, ideal, or hyperideal vertices. The surrounding discussion attributes related forms to earlier work, but the supplied text gives no resolution status.

Sources & referencesView supporting material

Primary source

Giulio Belletti and Tian Yang, “Asymptotics of quantum 6j-symbols and generalized hyperbolic tetrahedra”, arXiv:2308.13864 (2023).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.