A basic $n$ dimensional representation of Artin braid group $B_n$, and a general Burau representation

Abstract: Burau representation of the Artin braid group remains as one of the very important representations for the braid group. Partly, because of its connections to the Alexander polynomial which is one of the first and most useful invariants for knots and links. In the present work, we show that interesting representations of braid group could be achieved using a simple and intuitive approach, where we simply analyse the path of strands in a braid and encode the over-crossings, under-crossings or no-crossings into some parameters. More precisely, at each crossing, where, for example, the strand $i$ crosses over the strand $i+1$ we assign $\mathbf{t}$ to the {\bf t}op strand and $\mathbf{b}$ to the {\bf b}ottom strand. We consider the parameter $\mathbf{t}$ as a {\it relative weight} given to strand $i$ relative to $i+1$, hence the position $i\ i+1$ for $\mathbf{t}$ in the matrix representation. Similarly, the parameter $\mathbf{b}$ is a {\it relative weight} given to strand $i+1$ relative to $i$, hence the position $i+1\ i$ for $\mathbf{b}$ in the matrix representation. We show this simple {\it path analyzing approach} leads us to an interesting simple representation. Next, we show that following the same intuitive approach, only by introducing an additional parameter, we can greatly improve the representation into the one with much smaller kernel. This more general representation includes the unreduced Burau representation, as a special case. Our new {\it path analyzing approach} has the advantage that it applies a very simple and intuitive method capturing the fundamental interactions of the strands in a braid. In this approach we intuitively follow each strand in a braid and create a {\it history} for the strand as it interacts with other strands via over-crossings, under-crossings or no-crossings. This, directly, leads us to the desired representations.