Upper bounds for the Lagrangian cobordism relation on Legendrian links

TL;DR

This paper establishes upper bounds for the Lagrangian cobordism relation among Legendrian links in tight contact 3-manifolds, introducing a minimal genus concept and constructing explicit cobordisms.

Contribution

It provides the first known upper bounds for collections of Legendrian links under Lagrangian cobordism and defines a minimal genus measure between links.

Findings

Existence of an upper bound for finite Legendrian link collections.

Construction of explicit exact Lagrangian cobordisms.

Definition of a minimal Lagrangian genus between links.

Abstract

Lagrangian cobordism induces a preorder on the set of Legendrian links in any contact 3-manifold. We show that any finite collection of null-homologous Legendrian links in a tight contact 3-manifold with a common rotation number has an upper bound with respect to the preorder. In particular, we construct an exact Lagrangian cobordism from each element of the collection to a common Legendrian link. This construction allows us to define a notion of minimal Lagrangian genus between any two null-homologous Legendrian links with a common rotation number.