Lagrangian spheres and Dehn twists

In this paper we answer a question raised in [8] as to whether, at least in a simple situation, symplectic Dehn twists provide the only method of constructing symplectically knotted Lagrangian spheres in a symplectic 4-manifold. We first recall the arrangement from [8]. Let W be the Stein manifold formed by adding to the unit cotangent bundle of the two-sphere, T S, a single 2-handle along the Legendrian curve in a single fiber of the boundary. As a Stein manifold it carries a symplectic structure which has a conformally expanding vector field whose flow exists for all time. The symplectic structure is the Kähler form associated to a plurisubharmonic exhaustion function and all such forms are equivalent up to symplectomorphism (see [4]). Alternatively W can be realized as the plumbing of two copies of T S. Namely we take two copies of T S and identify the cotangent fibers projecting to a disk D in S with a product D × E in each copy. We then identify the two copies of D × E, preserving the product structure but reversing the factors. The resulting symplectic manifold W has two Lagrangian spheres L1 and L2 coming from the zero-sections in the T S (or, in the previous description, the original zero-section and the stable manifold of the index 2 critical point in the added handle). The following was proven in [8].