In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S2 or ℝℙ2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.
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Acknowledgments The authors warmly thank Selman Akbulut, Josef Dorfmeister, Ronald Fintushel, Robert Gompf, Dusa McDuff, and Leonid Polterovich for their interest in this work and many helpful correspondences. Particular thanks are due to Dusa McDuff for generously sharing early versions of her paper  with us, which plays a key role in our arguments. We would also like to thank the anonymous referee for valuable comments, suggestions, clarifications, and pointing us to the fact that Corollary 1.2(1) leads to a description of the Hamiltonian isotopy classes of Lagrangian spheres in a compact symplectic manifold where there are Hamiltonian knotted Lagrangian spheres. Matthew Strom Borman was partially supported by NSF-grant DMS 1006610; Tian-Jun Li and Weiwei Wu were supported by NSF-grant DMS 0244663.
- Lagrangian knots
- Rational manifolds
- Symplectic ball packing
- Symplectic cutting
- Symplectic manifolds