Constructions and Isotopies of High-Dimensional Legendrian Spheres

TL;DR

This paper investigates methods to construct Legendrian spheres in high-dimensional contact manifolds, demonstrating their isotopy to the Legendrian unknot and generalizing previous results.

Contribution

It introduces and compares three constructions of Legendrian spheres, proving their isotopy to the standard Legendrian unknot across all dimensions.

Findings

All three constructions produce Legendrian spheres isotopic to the unknot.

Ekholm's doubling procedure yields the standard Legendrian unknot.

The constructions work in any contact manifold, extending previous results.

Abstract

We explore the construction of Legendrian spheres in contact manifolds of any dimension. Two constructions involving open books work in any contact manifold, while one introduced by Ekholm works only in $\mathbb{R}^{2n+1}$. We show that these three constructions are isotopic to the Legendrian unknot, thus recovering and generalising a result of Courte and Ekholm, that shows Ekholm's doubling procedure produces the standard Legendrian unknot.