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

Abstract

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.

33 pages, 12 figures. arXiv admin note: text overlap with arXiv:2008.05045

References in corpus (3)

• On the Volume Conjecture for hyperbolic Dehn-filled $3$-manifolds along the figure-eight knot
• Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II
• A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifolds

Cited by in corpus (4)

• Relative Reshetikhin-Turaev invariants, hyperbolic cone metrics and discrete Fourier transforms II
• A relative version of the Turaev-Viro invariants and the volume of hyperbolic polyhedral 3-manifolds
• Computation of leading coefficients in asymptotics of relative quantum invariants
• On the asymptotic expansion for the relative Reshetikhin-Turaev invariants of fundamental shadow link pairs