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Unlinking and unknottedness of monotone Lagrangian submanifolds

Georgios Dimitroglou Rizell and Jonathan David Evans

Geometry & Topology 18 (2014) 997–1034

Under certain topological assumptions, we show that two monotone Lagrangian submanifolds embedded in the standard symplectic vector space with the same monotonicity constant cannot link one another and that, individually, their smooth knot type is determined entirely by the homotopy theoretic data which classifies the underlying Lagrangian immersion. The topological assumptions are satisfied by a large class of manifolds which are realised as monotone Lagrangians, including tori. After some additional homotopy theoretic calculations, we deduce that all monotone Lagrangian tori in the symplectic vector space of odd complex dimension at least five are smoothly isotopic.

Lagrangian submanifold, symplectic manifold, monotone, torus, knot
Mathematical Subject Classification 2010
Primary: 53D12
Received: 29 November 2012
Revised: 16 September 2013
Accepted: 16 October 2013
Published: 7 April 2014
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Simon Donaldson
Georgios Dimitroglou Rizell
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud
Bâtiment 425
F-91405 Orsay
Jonathan David Evans
Department of Mathematics
University College London
Gower Street
London WC1E 6BT