6j symbols for U_q(sl_2) and non-Euclidean tetrahedra

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 cas…

A formula with volumes of five tetrahedra and discrete curvature

Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the product of the two remaining tetrahedron volumes in its r.h.s., as well as the derivative of the "discrete curvature" which arises when we slightly deform our euclidean space.

量子$6j$-符号的渐近性与广义双曲四面体

Asymptotics of quantum $6j$-symbols and generalized hyperbolic tetrahedra

该研究在Turaev-Viro不变量的容许性条件下，揭示量子$6j$-符号的6元组对应四类几何结构的二面角，证明多数情形下其指数增长率等于广义双曲四面体体积，且体积可负导致序列指数衰减，填补了相关认知空白。

Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis

The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetrie…

Asymptotics of quantum $6j$ symbols

Asymptotics of quantum $6j$ symbols corresponding to a hyperbolic tetrahedra is investigated and the first two leading terms are determined for the case that the tetrahedron has a ideal or ultra-ideal vertex. These terms are given by the volume and the determinant of the Gram matrix of the tetrahedron. A relation to the volume conjecture of the Turaev-Viro invariant is also discussed.

Classical 6j-symbols and the tetrahedron

A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are thes…