关于辫群的循环运算

翻开书页的操作

Operations on open book foliations

研究了b弧叶理的变化和开卷叶理的交换运动，推广了辫状叶理理论中的相应运算。我们还定义了一个旁路移动作为本田的旁路附件操作的模拟。作为应用，我们研究了开卷叶理在开卷稳定条件下的变化规律。我们还推广了Birman-Menasco的分裂/复合辫子定理：分裂的封闭辫子代表（分别为）。复合）链接可以被转换为拆分（分别为复合）通过应用交换移动多次闭合编织。

Garside群中共轭问题的一种新方法

A New Approach to the Conjugacy Problem in Garside Groups

循环运算赋予Garside群$G$的任意元素$x$的超顶点集$S_x$有向图$Γ_x$的结构。我们建立了由$Γ_x$的电路组成的$S_x$的子集$U_x$可以用来代替$S_x$来决定$G$中的$x$的共轭性，从而更快更实用地解决Garside群的共轭性问题。此外，我们提出了一种概率方法的共轭搜索问题的Garside群。这些结果可能会对最近提出的基于辫群中共轭（搜索）问题的安全性产生影响。

带缆辫子及其交叉矩阵

Cabled braids and their crossing matrices

该研究针对辫子的加缆操作，刻画可约纯辫子的交叉矩阵，利用加缆构造辫子的增强共轭不变量。

Garside群的共轭性II：超顶点集的结构

Conjugacy in Garside groups II: Structure of the ultra summit set

本文是研究Garside群上的共轭判定问题和共轭搜索问题的系列文章的第二篇。Garside群G中元素X的超顶点集USS（X）是G中元素的有限集，它是G中X的共轭类的完全不变量。如果人们希望找到USS（X）大小的界限，一个根本问题就是了解它的结构。本文引入USS（X）中元素的两种新的运算，称为“部分循环”和“部分扭去循环”，并证明了：如果Y和Z属于USS（X），则Y和Z由部分循环和部分扭去循环序列相关联.这些运算是理解由第二作者的论文中的凸性定理所得到的最小简单元的一种具体方法。利用部分循环和部分扭去循环，研究了由USS（X）确定的有向图Γ X的结构，证明了Γ X可分解为“黑”和“灰”子图.在本系列的第一篇文章中，有一些与该程序相关的应用程序，用于在编织物的情况下找到CDP/CSP的多项式解。一个不同的应用是给出一个新的算法来解决CDP/CSP在Garside组这是比所有其他已知的算法，即使其理论复杂性是相同的，由第二作者。也有应用到理论的约化群。

The planar pure braid group is a diagram group

A planar pure braid consists of $n$ descending smooth arcs, each connecting a point on one horizontal line $\ell_{1}$ to a point on a horizontal line $\ell_{2}$, which is required to be directly below the first point. Two arcs are allowed to cross, but no threefold intersections are allowed. The set $Γ_{n}$ of all planar pure braids on $n$ strands is a group with respect to a natural stacking operation. We show that $Γ_{n}$ is always a diagram group, in the sense of Guba and Sapir. A number of consequences follow, including biautomaticity and bi-orderability of the groups $Γ_{n}$. Moreover, ea…

Garside群中的循环滑动运算

The cyclic sliding operation in Garside groups

我们提出了一个新的操作要执行的元素在Garside组，称为循环滑动，这是介绍，以取代众所周知的循环和decycling操作。循环滑动似乎是一个更自然的选择，简化了Garside群中共轭的算法，并具有更好的理论性质。我们表明，特别是，如果一个超级首脑会议的元素共轭是“刚性”（即，具有一定的特别简单的结构），然后通过共轭的正元素获得这样一个刚性共轭的最佳方式是由迭代循环滑动。

A theory of orbit braids

This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homot…

On the second homology of planar graph braid groups

We show that the second homology of the configuration spaces of a planar graph is generated under the operations of embedding, disjoint union, and edge stabilization by three atomic graphs: the cycle graph with one edge, the star graph with three edges, and the theta graph with four edges. We give an example of a non-planar graph for which this statement is false.

Formal descriptions of Turaev's loop operations

Using intersection and self-intersection of loops, Turaev introduced in the seventies two fundamental operations on the algebra $\mathbb{Q}[π]$ of the fundamental group $π$ of a surface with boundary. The first operation is binary and measures the intersection of two oriented based curves on the surface, while the second operation is unary and computes the self-intersection of an oriented based curve. It is already known that Turaev's intersection pairing has an algebraic description when the group algebra $\mathbb{Q}[π]$ is completed with respect to powers of its augmentation ideal and is app…