Every non-monotone toric symplectic four-manifold admits a Hamiltonian diffeomorphism and a Lagrangian torus that never intersects its own image under iteration.
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abstract
We study piecewise linear knot diagrams in the base of almost toric fibrations of symplectic four-manifolds. These diagrams translate to deformations of the almost toric fibration. We give several applications to symplectic topology, among them a proof of a conjecture by Symington, simpler counterexamples to Lagrangian Poincar\'e recurrence in dimension four, the calculation of the displacement energy for many fibres of toric moment maps, and an elementary recipe for building and distinguishing Lagrangian torus knots.
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More counterexamples to Lagrangian Poincar\'e recurrence in dimension four
Every non-monotone toric symplectic four-manifold admits a Hamiltonian diffeomorphism and a Lagrangian torus that never intersects its own image under iteration.