The quantum $\mathfrak{sl}(n,\mathbb{C})$ representation theory and its applications

Volume and Homology for Hyperbolic $3$-Orbifolds, I

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold whose singular set is a link, and such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that if the underlying manifold $|{\mathfrak M}|$ is irreducible, and $|{\mathfrak M}|$ is irreducible for every two-sheeted (orbifold) covering $\widetilde{\mathfrak M}$ of ${\mathfrak M}$, and if ${\rm vol} {\mathfrak M}\le1.72$, then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 15$. Furthermore, if ${\rm vol} {\mathfrak M}\le1.22$ then $\dim H_1({\mathfrak M};{\mathbb Z}_2)\le 11$, and if ${\rm vol} {\mathfrak M}\le0.…

带尖点的三维流形的$\boldsymbol{\rm U}_{q\tilde q}\frak{sl}(2;\boldsymbol{\rm R})$ Turaev-Viro不变量

$\mathrm {U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$ Turaev-Viro invariants for cusped $3$-manifolds

该研究拓展了双曲三维流形的Turaev-Viro不变量，构造出带尖点三维流形的$\boldsymbol{\rm U}_{q\tilde q}\frak{sl}(2;\boldsymbol{\rm R})$型不变量，并证明其衰减率由流形双曲体积决定。

Volume and Homology for Hyperbolic 3-Orbifolds

Let ${\mathfrak M}$ be a closed, orientable, hyperbolic 3-orbifold such that $π_1({\mathfrak M})$ contains no hyperbolic triangle group. We show that strict upper bounds of 0.07625, 0.1525 and 0.22875 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 23, 43 and 79 for $\dim H_1({\mathfrak M};{\mathbb F}_2 )$. Stronger results hold if we assume that the singular set $Σ$ is a link; specifically, under this assumption, strict upper bounds of 0.305, 0.4575, 0.61, 0.7625 and 0.915 for ${\rm vol}\ {\mathfrak M}$ imply respective upper bounds of 7, 13, 14, 28 and 29 for ${\rm dim}\ H_1(…

从U_qsl(2;R)的模双的6j符号出发的Turaev-Viro不变量

Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$

本文通过U_qsl(2;R)的模双的6j符号，定义了超几何3流形的Turaev-Viro型不变量，并证明其衰减率与双曲体积和adjoint扭转Reidemeister torsion相关。