Torsion in linearized contact homology for Legendrian knots

TL;DR

This paper demonstrates the existence of Legendrian knots with torsion in their linearized contact homology over integers and explores implications for augmentations and Lagrangian fillings.

Contribution

It provides the first examples of Legendrian knots with torsion in their linearized contact homology over , and shows that some augmentations are not induced by exact Lagrangian fillings.

Findings

Existence of Legendrian knots with torsion in their linearized contact homology over .

Not all augmentations over  are induced by exact Lagrangian fillings.

Implications for the relationship between augmentations and Lagrangian fillings.

Abstract

We present examples of Legendrian knots in $\mathbb{R}^3$ that have linearized Legendrian contact homology over $\mathbb{Z}$ containing torsion. As a consequence, we show that there exist augmentations of Legendrian knots over $\mathbb{Z}$ that are not induced by exact Lagrangian fillings, even though their mod $2$ reductions are.