Genus-zero fillings in the stabilization–satellite construction

Determine whether the construction that produces Legendrian knots Λ′± via sufficiently many positive and negative stabilizations followed by Legendrian satellite operations (used to obtain non-regular Lagrangian concordances between Lagrangian fillable Legendrian knots in the standard contact 3-sphere) can be carried out so that the resulting Legendrian knots admit exact Lagrangian fillings of genus zero.

Background

The paper constructs non-regular Lagrangian concordances between Lagrangian fillable (hence augmentable and non-stabilized) Legendrian knots by starting from a decomposable Lagrangian concordance and then applying a sequence of stabilizations and Legendrian satellite operations. The resulting Legendrian ends Λ′± are fillable.

In the present implementation of the method, the authors only obtain fillings of positive genus for Λ′± and explicitly note uncertainty about whether genus-zero fillings (i.e., Lagrangian disks) can be achieved within the same framework. Establishing this would refine the scope of the construction and clarify whether their approach can realize disk fillings while preserving the non-regularity of the constructed concordances.