辫群的结点矩阵表示法

Node Matrix Representation of Braid Group

利用矩阵乘法规则与辫子乘法中图形的上下对应连接有类似之处这一特点,提出一种用结点矩阵表示辫子的方法.经过标记结点,确定矩阵中非零元素的位置以及非零元素表达式3个步骤即可求出一个辫子的结点矩阵表达式.该方法能够解决共轭等辫群中的难解问题.与已有的Burau表示法相比,它在与图形的转化以及矩阵求逆等方面有着一定的优越性.

Based on the point that there is something in common between the rules of matrix multiplication and the figure connection in braid multiplication, this paper gave a new representation of braid group called node matrix representation. By signing the nodes, confirming the location of the nonzero elements in the matrix, and calculating the expression of the nonzero elements, the node matrix expression of a braid can be calculated. This method can also resolve some difficult problems in braid group like conjugacy problem. Compared with Burau representation, node matrix representation has advantages in transforming with figures and calculating the contradiction matrix.