Lagrangian concordance is not a symmetric relation

TL;DR

This paper provides an explicit example demonstrating that Lagrangian concordance between Legendrian knots is not a symmetric relation, using contact homology and augmentation categories.

Contribution

It constructs a specific Legendrian knot with a one-way Lagrangian concordance, proving non-symmetry of the relation.

Findings

Lagrangian concordance from the trivial knot to a non-trivial knot exists.

No concordance exists in the reverse direction for the same pair.

The non-symmetry is established via Legendrian contact homology maps.

Abstract

We provide an explicit example of a non trivial Legendrian knot $\Lambda$ such that there exists a Lagrangian concordance from $\Lambda_0$ to $\Lambda$ where $\Lambda_0$ is the trivial Legendrian knot. We then use the map induced in Legendrian contact homology by a concordance and the augmentation category of $\Lambda$ to show that no Lagrangian concordance exists in the other direction. This proves that the relation of Lagrangian concordance is not symmetric.