Discrete Fourier transforms, quantum $6j$-symbols and deeply truncated tetrahedra

The asymptotic behavior of quantum $6j$-symbols is closely related to the volume of truncated hyperideal tetrahedra\,\cite{C}, and plays a central role in understanding the asymptotics of the Turaev-Viro invariants of $3$-manifolds. In this paper, we propose a conjecture relating the asymptotics of the discrete Fourier transforms of quantum $6j$-symbols on one hand, and the volume of deeply truncated tetrahedra of various types on the other. As supporting evidence, we prove the conjecture in the case that the dihedral angles are sufficiently small, and provide numerical calculations in the case that the dihedral angles are relatively big. A key observation is a relationship between quantum $6j$-symbols and the co-volume function of deeply truncated tetrahedra, which is of interest in its own right. More ambitiously, we extend the conjecture to the discrete Fourier transforms of the Yokota invariants of planar graphs and volume of deeply truncated polyhedra, and provide supporting evidence.