A Garside-theoretic approach to the reducibility problem in braid groups
A Garside-theoretic approach to the reducibility problem in braid groups
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辫子群中可还原性问题的加赛德理论方法
DOI：
10.1016/j.jalgebra.2008.03.033
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
Sangjin Lee
中科院分区：
文献类型：
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作者：
Eonkyung Lee;Sangjin Lee
设Dn表示复平面中的n-穿孔圆盘，其中穿孔在真实的轴上。一个n-辫子α称为可约的，如果在Dn中存在一个本质曲线系C，称为α的约化系，使得α*C=C，其中α*C表示辫子α对曲线系C的作用。在Dnis中的曲线系统C被称为标准的，如果它的每一个组成部分都与以真实的轴为中心的圆形是同位素的。本文研究了将一个曲线系转化为一个标准曲线系的辫子的特征，进而研究了可约辫子的共轭类的特征。对于Dn中的本质曲线系C，定义C的扩张子St（C）={P∈Bn+：P <$C是标准的}，并证明St（C）是Bn+的子格.特别地，在St（C）中存在唯一的极小元。利用可约辫的极小元和可约辫的规范归约系统，定义了可约辫的最外分支，并证明了对于最外分支比全辫（包括分裂辫）简单的可约辫，其超顶点集的每个元素都有一个标准归约系统.这意味着，对于这样的辫子，找到一个约简系统就像找到一个超顶点集的单个元素一样容易。
Let Dndenote the n-punctured disk in the complex plane, where the punctures are on the real axis. An n-braid α is said to be reducible if there exists an essential curve system C in Dn, called a reduction system of α, such that α*C=C where α*C denotes the action of the braid α on the curve system C. A curve system C in Dnis said to be standard if each of its components is isotopic to a round circle centered at the real axis. In this paper, we study the characteristics of the braids sending a curve system to a standard curve system, and then the characteristics of the conjugacy classes of reducible braids. For an essential curve system C in Dn, we define the standardizer of C as St(C)={P∈Bn+:P∗C is standard} and show that St(C) is a sublattice of Bn+. In particular, there exists a unique minimal element in St(C). Exploiting the minimal elements of standardizers together with canonical reduction systems of reducible braids, we define the outermost component of reducible braids, and then show that, for the reducible braids whose outermost component is simpler than the whole braid (including split braids), each element of its ultra summit set has a standard reduction system. This implies that, for such braids, finding a reduction system is as easy as finding a single element of the ultra summit set.