An SL(2,C) Algebro-Geometric Invariant of Knots

Rigidity of moduli spaces and algebro-geometric constructions

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and holomorphic aspects, and so they also provide motivation for natural problems in geometric group theory and topology.

映射类群的代数几何子群

Algebro-geometric subgroups of mapping class groups

本文通过约束映射类群的代数几何子群（即光滑曲线模空间上复代数映射下曲线基本群的像），证明当亏格至少为3时，闭曲面映射类群的无限秩有限秩酉表示在代数几何子群上的限制必为无限维，并由此推出大多数此类紧化的万有覆盖是Stein流形。

Geometric limits of knot complements, II: Graphs determined by their complements

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.

从代数几何角度看不同性模空间的拓扑

An algebro-geometric perspective on the topology of moduli spaces of differentials

本文从代数几何角度综述不同性模空间的拓扑结构、已知结果及开放问题，探讨其与其他领域的联系。

扭结理论的几何深度学习方法

Geometric deep learning approach to knot theory

本文提出一种将扭结转化为图并利用图神经网络预测扭结不变量的几何深度学习方法，展示了该方法具有较高的泛化能力。

A geometric foundation of virtual knot theory

Virtual knots are defined diagrammatically as a collection of figures, called virtual knot diagrams, that are considered equivalent up to finite sequences of extended Reidemeister moves. By contrast, knots in $\mathbb{R}^3$ can be defined geometrically. They are the points of a space $\mathbb{K}$ of knots. The knot space has a topology so that equivalent knots lie in the same path component. The aim of this paper is to use sheaf theory to obtain a fully geometric model for virtual knots. The geometric model formalizes the intuitive notion that a virtual knot is an actual knot residing in a var…