Unknotting Lagrangian $\mathrm{S}^1\times\mathrm{S}^{n-1}$ in $\mathbb{R}^{2n}$

TL;DR

This paper classifies Lagrangian embeddings of the product of a circle and an (n-1)-sphere into 2n-dimensional Euclidean space up to smooth isotopy for all dimensions n ≥ 3.

Contribution

It provides a complete classification of these embeddings, advancing understanding of Lagrangian submanifolds in symplectic topology.

Findings

Classified all Lagrangian $ ext{S}^1 imes ext{S}^{n-1}$ embeddings in $ ext{R}^{2n}$ for n ≥ 3

Established isotopy equivalence classes for these embeddings

Enhanced the understanding of Lagrangian embedding structures in high-dimensional Euclidean spaces

Abstract

Lagrangian embeddings $\mathrm{S}^1\times\mathrm{S}^{n-1}\hookrightarrow\mathbb{R}^{2n}$ are classified up to smooth isotopy for all $n\ge 3$.