Constructions of Lagrangian cobordisms

TL;DR

This paper explores the construction and properties of Lagrangian cobordisms between Legendrian knots, introducing combinatorial and geometric methods, and surveying existing results related to their decomposability and nondecomposability.

Contribution

It introduces new combinatorial and geometric techniques for constructing Lagrangian cobordisms and surveys existing results on their decomposability.

Findings

Describes elementary building blocks for Lagrangian cobordisms.

Introduces satellite-based geometric methods for construction.

Reviews results suggesting the existence of nondecomposable cobordisms.

Abstract

Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combinatorial building blocks as well as some geometric methods, involving the theory of satellites, to construct Lagrangian cobordisms. We will then survey some known results, derived through Heegaard Floer Homology and contact surgery, that may provide a pathway to proving the existence of nondecomposable (nonribbon) Lagrangian cobordisms.