Non-looseness of boundaries of Legendrian ribbons

TL;DR

The paper investigates conditions under which the boundary of a Legendrian ribbon in an overtwisted contact 3-manifold is non-loose, introducing the 'Tight Reattachment Property' as a sufficient criterion.

Contribution

It defines the 'Tight Reattachment Property' for Legendrian graphs and demonstrates its role in ensuring non-looseness of ribbon boundaries, advancing understanding of Legendrian graph boundaries.

Findings

The 'Tight Reattachment Property' implies non-looseness of ribbon boundaries.

Not all boundaries of Legendrian ribbons are non-loose without this property.

Examples and constructions illustrate the applicability of the property.

Abstract

Every null-homologous link in an oriented 3-manifold is isotopic to the boundary of a ribbon of a Legendrian graph for any overtwisted contact structure. However this is not the case if the boundary is required to be non-loose. Here, we define the `Tight Reattachment Property' for a Legendrian graph and show that it implies the boundary of its ribbon is non-loose. We also discuss the applicability of this property and examine examples and constructions of Legendrian graphs with this property.