We provide an explicit example of a non trivial Legendrian knot such that there exists a Lagrangian concordance from to where 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 to show that no Lagrangian concordance exists in the other direction. This proves that the relation of Lagrangian concordance is not symmetric.
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Baptiste Chantraine, Lagrangian concordance is not a symmetric relation. Quantum Topol. 6 (2015), no. 3, pp. 451–474DOI 10.4171/QT/68