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

TL;DR

This paper develops new Legendrian homology invariants for sutured Legendrians with boundary, extending Floer theory techniques to define and relate cylindrical, wrapped, and Chekanov-Eliashberg invariants, with applications to classifying local 2-braids.

Contribution

It introduces a sutured Legendrian framework with invariants derived from Floer theory, including an exact sequence and a wrapped Chekanov-Eliashberg algebra, for Legendrians with boundary.

Findings

Invariants are preserved under boundary-fixed Legendrian isotopy.

The sutured Legendrian invariants distinguish certain manifold features.

Isotopic conormals imply braid equivalence in local 2-braids.

Abstract

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.