  We establish relations between both the classical and the dual Garside
structures of the braid group and the Burau representation. Using the classical
structure, we formulate a non-vanishing criterion for the Burau representation
of the 4-strand braid group. In the dual context, it is shown that the Burau
representation for arbitrary braid index is injective when restricted to the
set of \emph{simply-nested braids}.
