Non-looseness of boundaries of Legendrian ribbons

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.