辫群中的Hilden双陪集问题

纽结的桥位置与平台表示

Bridge Positions and Plat Presentations of Links

本文建立纽结的桥位置与平台表示的Hilden双陪集类的对应关系，将平台表示的代数问题转化为桥位置的几何问题，重新确立相关结果并探讨其在辫子平台闭包研究中的应用。

Mapping class groups of covers with boundary and braid group embeddings

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hil…

分支覆盖上的辫群作用与完全例外序列

Braid group actions on branched coverings and full exceptional sequences

本文通过将完全例外序列与圆盘上的分支覆盖相联系，利用Birman–Hilden理论和Hurwitz系统研究辫群作用，并构造了Bondal–Polishchuk传递性猜想的反例。

Braid structures in knot complements, handlebodies and 3-manifolds

We consider braids on $m+n$ strands, such that the first $m$ strands are trivially fixed. We denote the set of all such braids by $B_{m,n}$. Via concatenation $B_{m,n}$ acquires a group structure. The objective of this paper is to find a presentation for $B_{m,n}$ using the structure of its corresponding pure braid subgroup, $P_{m,n}$, and the fact that it is a subgroup of the classical Artin group $B_{m+n}$. Then we give an irredundant presentation for $B_{m,n}$. The paper concludes by showing that these braid groups or appropriate cosets of them are related to knots in handlebodies, in knot …

Classification of symmetry groups for planar n-body choreographies

Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry properties. In this paper, we give a systematic treatment of the symmetry aspect. In the first part we classify all possible symmetry groups of planar n-body, collision-free choreographies. These symmetry groups fall in to 2 infinite families and, if n is odd, three exceptional …

On exotic modular tensor categories

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the quantum double of the even sectors of the $E_6$ subfactor, and the other is the quantum double of the even sectors of the Haagerup subfactor. We cannot prove mathematically that the two TQFTs are indeed counterexamples because CSW TQFTs, while physically defined, are not yet mat…