  We consider knotted surfaces and their graphical description that include
2-dimensional braids and their chart description. We introduce a new
construction of surfaces in $D^2 \times I \times I$, called BMW surfaces, that
are described as the trace of deformations of tangles generated by generators
$g_i, g_i^{-1}, e_i$ of the BMW (Birman-Murakami-Wenzl) algebras, where $g_i$
and $g_i^{-1}$ are a standard generator of the $n$-braid group and its inverse,
and $e_i$ is a tangle consisting of a pair of "hooks". And we introduce a
notion of BMW charts that are graphs in $I \times I$ and show that a BMW
surface has a BMW chart description.
