Nonfillable Legendrian knots in the $3$–sphere

Abstract

If $\Lambda$ is a Legendrian knot in the standard contact $3$–sphere that bounds an orientable exact Lagrangian surface $\Sigma$ in the standard symplectic $4$–ball, then the genus of $\Sigma$ is equal to the slice genus of (the smooth knot underlying) $\Lambda$, the rotation number of $\Lambda$ is zero as well as the sum of the Thurston–Bennequin number of $\Lambda$ and the Euler characteristic of $\Sigma$, and moreover, the linearized contact homology of $\Lambda$ with respect to the augmentation induced by $\Sigma$ is isomorphic to the (singular) homology of $\Sigma$. It was asked by Ekholm, Honda and Kálmán (2016) whether the converse of this statement holds. We give a negative answer, providing a family of Legendrian knots with augmentations which are not induced by any exact Lagrangian filling although the associated linearized contact homology is isomorphic to the homology of the smooth surface of minimal genus in the $4$–ball bounding the knot.