Burau matrix of braid

Question

What is the definition of a Burau matrix of a braid? Where can I find a definition?

Answer 1

I personally like Birman's exposition in her book Braids, Links and Mapping Class Groups which gives a very topological introduction to the subject.

I also know that Daan Krammer gives a rather nice algebraic approach in a set of online lecture notes. He also has a list of exercises associated to the notes.

Answer 2

The other answer seems to be correct, but in the interest of providing a non-link-only answer, I'll give the definition here.

The Burau matrix of a braid is the matrix representing the braid in the Burau representation, so it's a matrix over $\mathbb Z [t, t^{-1}]$, i.e. the entries of the matrix are Laurent polynomials in the variable $t$ with integer coefficients. The Burau representation is a representation (i.e. group homomorphism) $\psi_n: B_n \rightarrow \operatorname{GL}_n(\mathbb Z [t, t^{-1}])$. As with any group homomorphism, it's sufficient to define it on a generating set and check that the resulting map is well-defined.

We define in block form as $$\psi_n : \sigma_i \mapsto \left( \begin{array}{ccc} I_{i-1} & 0 & 0 & 0 \\ 0 & 1-t & t & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & I_{n-i-1} \end{array} \right) ,$$ where $I_m$ is the $m \times m$ identity matrix, and $i$ ranges from $1$ to $n-1$. For this to be a valid homomorphism, we need to check that these matrices are invetible and that the braid relations are preserved, i.e. that $\psi_n(\sigma_i) \psi_n(\sigma_j) = \psi_n(\sigma_j) \psi_n(\sigma_i)$ for $|i-j| \ge 2$, and that $\psi_n(\sigma_i) \psi_n(\sigma_{i+1}) \psi_n(\sigma_i) = \psi_n(\sigma_{i+1}) \psi_n(\sigma_i) \psi_n(\sigma_{i+1})$ for $i=1, \ldots, n-2$. I'll exhibit the inverse matrix here, but leave it up to you to check that the computations work out. $$\psi_n(\sigma_i)^{-1} = \left( \begin{array}{ccc} I_{i-1} & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & t^{-1} & 1-t^{-1} & 0 \\ 0 & 0 & 0 & I_{n-i-1} \end{array} \right)$$

With that out of the way, for any braid $\gamma$, to compute its Burau matrix, just write $\gamma = \sigma_{i_1}^{e_1} \cdots \sigma_{i_m}^{e_m}$ as a word in the generators, and then you have $\psi_n(\gamma) = \psi_n (\sigma_{i_1})^{e_1} \cdots \psi_n (\sigma_{i_m})^{e_m}$, which is now easy to compute.

For some intuition, the case $t=1$ is good to look at. Setting $t=1$, we get a permutation matrix. This is the permutation matrix corresponding to the underlying permutation of the braid, i.e. the composite homomorphism $B_n \rightarrow S_n \rightarrow \operatorname{GL}_n(\mathbb Z)$ where $S_n$ is the symmetric group on $n$ letters. The Burau representation is thus 1-parameter deformation of the permutation representation.

The above is, of course, only one of several conventions, but it's the one I usually see chosen in the literature when one needs to do these calculations explicitly.