Constructions and Isotopies of High-Dimensional Legendrian Spheres

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.