On linking of Lagrangian tori in $\mathbb{R}^4$

TL;DR

This paper investigates the linking properties of Lagrangian tori in four-dimensional symplectic space, using holomorphic disk counts to establish linking results and extending previous work to non-monotone cases.

Contribution

It introduces new linking results for Lagrangian tori in , extending and strengthening prior work on monotone and non-monotone tori using enumerative holomorphic disk counts.

Findings

Any two Clifford tori are unlinked in a strong sense

Extended linking results to non-monotone Lagrangian tori in 

Strengthened previous conclusions on monotone tori linking

Abstract

We prove some results about linking of Lagrangian tori in the symplectic vector space $(\mathbb{R}^4, \omega)$. We show that certain enumerative counts of holomophic disks give useful information about linking. This enables us to prove, for example, that any two Clifford tori are unlinked in a strong sense. We extend work of Dimitroglou Rizell and Evans on linking of monotone Lagrangian tori to a class of non-monotone tori in $\mathbb{R}^4$ and also strengthen their conclusions in the monotone case in $\mathbb{R}^4$.