Representation of links by braids: A new algorithm
Representation of links by braids: A new algorithm
复制标题
用辫子表示链接：一种新算法
DOI：
10.1007/bf02566597
复制
发表时间：
1990
影响因子：
0.9
通讯作者：
P. Vogel
中科院分区：
文献类型：
--
作者：
P. Vogel
如果a是一个有n个分量的辫子，则a的闭包（表示为#）是通过用n条标准曲线将顶层的端点连接到底层的端点来构造的。这个过程产生一个有向链路~，它的交叉数与a相同。亚历山大[1]，[2]，[3]的一个经典结果指出，每个定向链都与一个闭辫线是同位素的。在他的证明亚历山大修改图的定向链接的一系列基本操作，以获得一个封闭的辫子。在这种变换过程中，“图像的几何形状”完全改变了。在亚历山大的算法的许多应用中，具有很少交叉点的链接产生具有大量交叉点的闭合辫。另一方面，许多代数不变量的链接首先定义的辫子。如果我们希望计算“小”链路L的这些不变量，则具有以下原理将非常有用：
If a is a braid with n components, the closure of a, denoted #, is constructed by connecting the endpoints at the top level to the bottom endpoints with n standard curves. This procedure yields an oriented link ~ having the same number of crossings as a. A classical result of Alexander [I], [2], [3] states that every oriented link is isotopic to a closed braid #. In his proof Alexander modifies the diagram of an oriented link by a sequence of elementary operations to obtain a closed braid. During this transformation the "geometry of the picture" is completely changed. In many applications of Alexander's algorithms links with few crossings yield closed braid with a large number of crossings. On the other hand many algebraic invariants of links are first defined on braids. If we wish to compute these invariants for a "small" link L, it will be very useful to have the following principle: