曲面微分同胚的量子不变量的广义Bonahon-Wong-Yang体积猜想 I：八结补空间

3-流形的量子迹映射

A Quantum trace map for 3-manifolds

本文为具有环面边界的3-流形定义了基于理想三角剖分的量子迹映射，将Bonahon-Wong和Lê的曲面量子迹映射推广到三维。

从 Dehn-Thurston 坐标看曲面辫代数到量子环面的嵌入

An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

本文利用辫代数在柄体辫模上的作用，构造了曲面 Kauffman 括号辫代数到局部化量子环面的嵌入，并借此研究了单位根处的表示，给出了 Bonahon-Wong 唯一性猜想的新证明及唯一表示的显式重构方法。

Triangular decomposition of skein algebras

By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.

A duality map for quantum cluster varieties from surfaces

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon and Wong.

源自带状态Skein代数的量子和乐链接不变量

Quantum holonomic link invariants derived from stated skein algebras

本文基于带状态skein代数表示理论为标架链环定义了不变量，推导了其规范不变性，揭示了与量子群及量子双曲几何构造的联系，并给出了R矩阵显式公式以计算不变量。

Quantum traces and embeddings of stated skein algebras into quantum tori

The stated skein algebra of a punctured bordered surface (or equivalently, a marked surface) is a generalization of the well-known Kauffman bracket skein algebra of unmarked surfaces and can be considered as an extension of the quantum special linear group $\mathcal{O}_{q^2}(SL_2)$ from a bigon to general surfaces. We show that the stated skein algebra of a punctured bordered surface with non-empty boundary can be embedded into quantum tori in two different ways. The first embedding can be considered as a quantization of the map expressing the trace of a closed curve in terms of the shear coor…

Quantum torus methods for Kauffman bracket skein modules

We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we …

考夫曼括号缠结算子与体积猜想

Kauffman bracket intertwiners and the volume conjecture

本文针对考夫曼括号缠结算子与体积猜想，通过代数嵌入显式计算闭环面缠结算子并证明其迹的上极限为零，还对负欧拉特征曲面的周期微分同胚提出并证明了相应极限为零的猜想。

$\mathrm{SL}_n(\mathbb{C})$ 的量子迹：$n=3$ 的情形

Quantum traces for $\mathrm{SL}_n(\mathbb{C})$: The case $n=3$

本文研究如何将Bonahon-Wong的SL2量子迹映射推广至SL3情形；作者利用Fock-Goncharov X坐标的q形变，在理想三角剖分下为加厚穿孔曲面中的标架有向链环构造Laurent多项式量子迹，并进一步提出一般SLn版本的不变量定义。