Obstructions to reversing Lagrangian surgery in Lagrangian fillings

TL;DR

This paper investigates the limitations of reversing Lagrangian surgery in Legendrian knot fillings, showing that certain immersed fillings cannot be obtained from simpler ones, and introduces obstructions based on augmentation counts.

Contribution

It establishes obstructions to reversing Lagrangian surgery by linking immersed fillings to Lagrangian cobordisms and augmentation counts.

Findings

Not all immersed, Maslov-zero, exact Lagrangian fillings can be obtained by reversing surgery.

A connection between immersed fillings with zero-action double points and embedded Lagrangian cobordisms is established.

Augmentation counts provide obstructions to the existence of certain Lagrangian cobordisms.

Abstract

Given an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-$0$, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-$0$, exact Lagrangian filling with genus $g \geq 1$ and $p$ double points can be obtained from such a Lagrangian surgery on a filling of genus $g-1$ with $p+1$ double points. To show this, we establish the connection between the existence of an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian $\Lambda$ that has $p$ double points with action $0$ and the existence of an embedded, Maslov-$0$, exact Lagrangian cobordism from $p$ copies of a Hopf link to $\Lambda$. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-$0$, exact Lagrangian cobordisms between Legendrian links.