Lagrangian concordance is not a symmetric relation

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 with maximal Thurston–Bennequin number. 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.