Legendrian knots and exact Lagrangian cobordisms

TL;DR

This paper develops new methods to construct and analyze exact Lagrangian cobordisms between Legendrian links, using Symplectic Field Theory invariants and combinatorial tools, revealing non-isotopic fillings.

Contribution

It introduces a gradient flow tree approach to compute DGA maps for exact Lagrangian cobordisms, providing a combinatorial framework for understanding their invariants.

Findings

Constructed exact Lagrangian cobordisms with cylindrical Legendrian ends.

Provided a combinatorial description of DGA maps for elementary cobordisms.

Discovered non-isotopic exact Lagrangian fillings of the same Legendrian link.

Abstract

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $\Lambda_+$ and $\Lambda_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $\Lambda_+$ to that of $\Lambda_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely combinatorial description of the cobordism map for elementary cobordisms, i.e., cobordisms that correspond to certain local modifications of Legendrian knots. As an application, we find exact Lagrangian surfaces that fill a fixed Legendrian link and are not isotopic through exact Lagrangian surfaces.