  Any link that is the closure of a positive braid has a natural Legendrian
representative. These were introduced in an earlier paper, where their
Chekanov--Eliashberg contact homology was also evaluated. In this paper we
re-phrase and improve that computation using a matrix representation. In
particular, we present a way of finding all augmentations of such Legendrians,
construct an augmentation which is also a ruling, and find surprising links to
LU-decompositions and Gr\"obner bases.
