Non-fillable augmentations of twist knots

TL;DR

This paper demonstrates the existence of Legendrian twist knot augmentations that cannot be realized by orientable Lagrangian fillings, using Floer theory and algebraic geometry techniques.

Contribution

It introduces new examples of non-fillable Legendrian knot augmentations and applies a Floer-theoretic approach to distinguish fillable from non-fillable cases.

Findings

Certain Legendrian twist knot augmentations are proven to be non-fillable.

A Floer-theoretic isomorphism with local coefficients is used to analyze augmentation varieties.

No algebraic torus corresponding to the fillings can exist for these augmentations.

Abstract

We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel-Ekholm-Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the first Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist.