Lagrangian spheres in Del Pezzo surfaces

TL;DR

This paper classifies Lagrangian spheres in symplectic Del Pezzo surfaces obtained from blow-ups of the complex projective plane in four or fewer points, revealing no Lagrangian knotting unlike the five-point case.

Contribution

It provides a classification of Lagrangian spheres up to isotopy in certain Del Pezzo surfaces and shows the absence of Lagrangian knotting in these cases.

Findings

Lagrangian spheres are classified up to isotopy in these surfaces.

No Lagrangian knotting occurs in the 4-or-fewer point blow-ups.

Contrasts with the 5-point blow-up case where knotting is present.

Abstract

Lagrangian spheres in the symplectic Del Pezzo surfaces arising as blow-ups of the complex projective plane in 4 or fewer points are classified up to Lagrangian isotopy. Unlike the case of the 5-point blow-up, there is no Lagrangian knotting.