  We extend the sutured framework to the case of Legendrians with boundary.
Using ideas from Lagrangian Floer theory, we define the cylindrical and the
wrapped sutured Legendrian homologies of a pair of sutured Legendrians. They
fit together into an exact sequence, and the exact triangle is invariant along
an Legendrian isotopy fixed at the boundary. For a single Legendrian, we also
define a wrapped version of its Chekanov-Eliashberg dga. Our main example of
sutured Legendrian is obtained via the unit conormal construction : a
submanifold $N \subset M$, such that $\partial N \subset \partial M$ , induces
a sutured Legendrian $\Lambda_N \subset ST^*M$, thus we get smooth invariants
of manifolds with boundary. As a simple application, we show that if the
conormals of two local 2-braids are isotopic (as Legendrians with fixed
boundary), then the braids are equivalent.
