# Bulky Hamiltonian isotopies of Lagrangian tori with applications

## TL;DR

This paper constructs special Lagrangian tori in four-dimensional balls that are Hamiltonian isotopic to the Clifford torus only in larger balls, leading to new symplectic embedding examples and characterizations of key symplectic objects.

## Contribution

It introduces monotone Lagrangian tori with unique Hamiltonian isotopy properties and applies them to create symplectically knotted embeddings and characterizations.

## Key findings

- Existence of Lagrangian tori isotopic to the Clifford torus only in larger balls.
- Construction of symplectically knotted embeddings of toric domains.
- Characterization of the Clifford torus via embedding properties.

## Abstract

We exhibit monotone Lagrangian tori inside the standard symplectic four-dimensional unit ball that become Hamiltonian isotopic to the Clifford torus, i.e.~the standard product torus, only when considered inside a strictly larger ball (they are not even symplectomorphic to a standard torus inside the unit ball). These tori are then used to construct new examples of symplectic embeddings of toric domains into the unit ball which are symplectically knotted in the sense of J.~Gutt and M.~Usher. We also give a characterisation of the Clifford torus inside the ball as well as the projective plane in terms of quantitative considerations; more specifically, we show that a torus is Hamiltonian isotopic to the Clifford torus whenever one can find a symplectic embedding of a sufficiently large ball in its complement.