Burau Representation

Gives a Matrix representation ${\hbox{\sf b}}_i$ of a Braid Group in terms of $(n-1)\times (n-1)$ Matrices. A always appears in the position.

$\displaystyle {\hbox{\sf b}}_1$	$\textstyle =$	$\displaystyle \left[\begin{array}{ccccc}-t & 0 & 0 & \cdots & 0 \\ -1 & 1 & 0 ... ...& \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 1 & \cdots & 1\end{array}\right]$	(1)
$\displaystyle {\hbox{\sf b}}_i$	$\textstyle =$	$\displaystyle \left[\begin{array}{cccccc}1 & \cdots & 0 & 0 & \cdots & 0 \\ \v... ... 0 & 0 & \ddots & \vdots \\ 0 & \cdots & 0 & 0 & \cdots & 1 \end{array}\right]$	(2)

$\begin{displaymath} {\hbox{\sf b}}_{n-1} = \left[{\matrix{1 & 0 & \cdots & 0 & 0... ... 0 & \cdots & 0 & -t \cr 0 & 0 & \cdots & 0 & -t \cr}}\right]. \end{displaymath}$

(3)

Let $\Psi$ be the Matrix Product of Braid Words, then

$\begin{displaymath} {\hbox{det}({\hbox{\sf I}}-\Psi)\over 1+t+\cdots + t^{n-1}} = \Delta_L, \end{displaymath}$

(4)

where $\Delta_L$ is the Alexander Polynomial and det is the Determinant.

References

Burau, W. ``Über Zopfgruppen und gleichsinnig verdrilte Verkettungen.'' Abh. Math. Sem. Hanischen Univ. 11, 171-178, 1936.

Jones, V. ``Hecke Algebra Representation of Braid Groups and Link Polynomials.'' Ann. Math. 126, 335-388, 1987.