\begin{definition}[Definition:Braid Group]
The braid group on $n$ strands is generated by $\sigma_1, \sigma_2, \ldots, \sigma_{n-1}$ and the following relations:
# $\sigma_i \sigma_j = \sigma_j\sigma_i, \forall i, j: |i-j| \ge 2$
# $\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \forall i \in \{ 1, 2, \ldots, n-2 \}$
\end{definition}