Wrapped sutured Legendrian homology and unit conormal of local 2-braids

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 $\mathrm{\partial }N\subset \mathrm{\partial }M$ , induces a sutured Legendrian ${\mathrm{\Lambda }}_N\subset S{T}^{\ast }M$, 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.