Lagrangian concordance of Legendrian knots

Abstract

In this article we define Lagrangian concordance of Legendrian knots, the analogue of smooth concordance of knots in the Legendrian category. In particular we study the relation of Lagrangian concordance under Legendrian isotopy. The focus is primarily on the algebraic aspects of the problem. We study the behavior of the classical invariants under this relation, namely the Thurston-Bennequin number and the rotation number, and we provide some examples of non-trivial Legendrian knots bounding Lagrangian surfaces in $D^4$. Using these examples, we are able to provide a new proof of the local Thom conjecture.Comment: 18 pages, 4 figures. v2: Minor corrections and a proof of conjecture 6.4 of version 1 (now Theorem 6.4). v3: Several substantial changes notably the proof of theorems 1.2 and 5.1, this is the version accepted for publication in "Algebraic and Geometric Topology" published in January 201

Obstructions to the Existence and Squeezing of Lagrangian Cobordisms

“…To prove Theorem 1.1, which has no extendibility hypothesis on the cobordism, we need the following geometric result, whose proof is a simple modification of [1,Theorem 6.4] with the writhe taking the place of the Thurston-Bennequin number. …”

A note on the front spinning construction

“…We first observe that (1) and (2) are straightforward and follow from the construction of Σ S m Λ. Then we prove (3). Note that L Σ S m Λ := Σ S m L Λ is diffeomorphic to L Λ × S m .…”

Some non-collarable slices of Lagrangian surfaces

“…In Section 2, we recall the previous results on the construction of Lagrangian cobordisms of Figure 1. We also recall the main obstruction to the existence of Lagrangian fillings from [1]. In Sections 3 and 4, we give the constructions of the two Lagrangians of Theorems 1.7 and 1.8 and prove that the slices are non-collarable.…”

“…If, under this identification, the slice L Y is collarable, then cutting along Y leads to two fillings of a Legendrian submanifold of Y . Such fillings were introduced and studied by the author in [1]. From there one could deduce some topological constraints on both L and the Legendrian slice.…”

“…The second one is the context of Lagrangian cobordisms, also introduced by the author in [1]. We recall here the definition of this notion.…”