Lagrangian concordance is not a symmetric relation
This paper is a natural continuation of the author's earlier work [Algebr. Geom. Topol. 10, No. 1, 63--85 (2010; Zbl 1203.57010)], where it was proved that Lagrangian cobordism does not induce a symmetric relation on Legendrian knots, and that an invertible Lagrangian cobordism must in fact be a concordance. The main result of this paper shows that the Legendrian representative \(\Lambda \subset (\mathbb{R}^3,\xi_{st})\) of the knot \(m(9_{46})\) is the positive end of a Lagrangian concordance from the unknot, but does not admit a Lagrangian concordance to the unknot. This shows that the relation of Lagrangian concordance is not symmetric, even in the simplest possible case. The proof combines two main ingredients to yield a contradiction. The first ingredient, stated as Theorem 1.2, is of independent interest, and asserts that any self-concordance of the unknot with maximal Thurston-Bennequin invariant is in fact symplectomorphic to the product concordance. The second ingredient is the computation of maps induced by Lagrangian concordances and the augmentation category of \(\Lambda\).
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