Unlinking and unknottedness of monotone Lagrangian submanifolds

TL;DR

This paper proves that under certain conditions, monotone Lagrangian submanifolds in symplectic vector spaces cannot link and their knot type is determined by homotopy data, leading to isotopy results for monotone Lagrangian tori.

Contribution

It establishes unlinking and unknottedness results for monotone Lagrangians, showing their knot type is dictated by homotopy data and proving isotopy of monotone Lagrangian tori in high dimensions.

Findings

Monotone Lagrangians with the same monotonicity constant cannot link.

Their smooth knot type is determined by homotopy data.

All monotone Lagrangian tori in certain dimensions are smoothly isotopic.

Abstract

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.