Generalized Long-Moody representations of braid groups

广义辫子的拟环面表示

Quasitoric representation of generalized braids

本文基于Fenn和Bartholomew的纽结理论语言定义广义辫子理论，计算其生成集后证明定向正规广义纽结是拟环面正规广义辫子的闭包，且后者构成正规广义辫子群的子群。

A representation of generalized braid group in classical braid group

We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type A_n, B_n, I_2(k). The injectivity question of this homomorphism (in these particular cases) is not yet settled. If this map is an injective homomorphism then several results will follow. For example it will follow that the Whitehead group, projective class group and the lower K-…

Katz-Long-Moody Hermite 型的一个闭式签名公式

A closed signature formula for the Katz-Long-Moody Hermitian form

本文针对 Katz-Long-Moody 构造给出的商表示上的不变 Hermite 型，提出了一个闭式签名公式，该公式基于输入局部单值、其有序乘积的本征角及卷积参数，解决了正定性问题，并应用于 Pochhammer 系统及 Hecke 和 Temperley-Lieb 特殊化。

广义勒夫谢茨数与环辫群

The generalized Lefschetz number and loop braid groups

本文通过引入环辫群和广义勒夫谢茨数，研究三维流形中同胚的固定点和周期点的拓扑动力学性质。

The Burau representation is not faithful for n = 5

The Burau representation is a natural action of the braid group B_n on the free Z[t,t^{-1}]-module of rank n-1. It is a longstanding open problem to determine for which values of n this representation is faithful. It is known to be faithful for n=3. Moody has shown that it is not faithful for n>8 and Long and Paton improved on Moody's techniques to bring this down to n>5. Their construction uses a simple closed curve on the 6-punctured disc with certain homological properties. In this paper we give such a curve on the 5-punctured disc, thus proving that the Burau representation is not faithful…

Ordering pure braid groups on closed surfaces

We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.

Families of superelliptic curves, complex braid groups and generalized Dehn twists

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We find that $E_n^d$ is the classifying space for the complex braid group of type $B(d,d,n)$ and we compute a big part of the integral homology of $E_n^d,$ including a complete calculation of the stable groups over finite fields by means of Poincarè series. The computation of the main …

广义J群、J辫群和Seifert链环群

Generalized J-groups, J-braid groups and Seifert link groups

本文研究了广义J群的性质，确定其有限生成条件，并展示了其与复反射群及J辫群torsion商的关系。

Minimal generating sets and the abelianization for the quasitoric braid group

A toric braid is a braid whose closure is a torus link in $\mathbb{R}^3$. Manturov generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization.

复辫子群的抛物子群：剩余情形

Parabolic subgroups of complex braid groups: the remaining case

本文利用复辫子群 B(G31) 的 Garside 广群结构，证明 Marin 和 González-Meneses 关于抛物子群的主要定理，补全了此前唯一遗留情形。