Relation between immersed cobordism maps and augmentation varieties

Determine how the algebraic subset of augmentations induced by the immersed Lagrangian cobordism associated with a Legendrian clasp move relates to the full augmentation variety of the Legendrian after the clasp move and to the augmentation variety before the clasp move.

Background

For a clasp move between Legendrians, the associated immersed Lagrangian cobordism defines an algebraic subset of the augmentation variety of the post-move Legendrian. The paper establishes the full change in augmentations for Legendrians of dimension at least two, but does not identify the relationship between this subset and the augmentation varieties on either side of the move.

References

It is unknown to the author how the inclusion $I_\Sigma\subset Aug(\Lambda_\mu)$ and the variety $Aug(\Lambda_{-\mu})$ relate to each other.

— Floor-crossing for Legendrian Clasp Move  (2609.01381 - Strakoš, 1 Sep 2026) in Section 1, Introduction

Our methods rely on a~certain genericity condition for $J$-holomorphic curves (see the~proof of Proposition~\ref{prop: line segment} and Proposition 4.10) that fails in the~case of Legendrian knots, and so the~full effect of the~clasp move on the~augmentation variety remains open for $dim \,\Lambda_\mu=1$.

— Floor-crossing for Legendrian Clasp Move  (2609.01381 - Strakoš, 1 Sep 2026) in Section 1, Introduction