# Nonfillable Legendrian knots in the 3-sphere

## TL;DR

This paper investigates the relationship between Legendrian knots and Lagrangian fillings in the 3-sphere, providing counterexamples to a previously posed conjecture about their correspondence.

## Contribution

It constructs Legendrian knots with augmentations not arising from any exact Lagrangian filling, disproving a conjecture about their equivalence.

## Key findings

- Counterexamples to the conjecture about Legendrian knots and Lagrangian fillings.
- Augmentations can exist without corresponding Lagrangian fillings.
- Linearized contact homology can match the homology of minimal genus surfaces.

## Abstract

If a Legendrian knot $\Lambda$ in the standard contact 3-sphere 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 sum of the Thurston-Bennequin number of L and the Euler characteristic of $\Sigma$ is zero as well as the rotation number of $\Lambda$, 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 in arXiv:1212.1519 whether the converse of this statement is true. We give a negative answer to this question by providing a family of Legendrian knots with augmentations 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.