Floor-crossing for Legendrian Clasp Move

Abstract

We investigate the effect of the clasp move of Legendrian submanifolds of dimension at least two on the moduli spaces of pseudo-holomorphic curves that contribute to the differential of the Chekanov-Eliashberg algebra. As an application, we compute the differential of Chekanov-Eliashberg algebra of twist spheres $Λ_k$ of dimension at least two. Moreover, we construct an infinite family of Legendrians $\mathcal{T}(Λ,k)$ from any horizontally displaceable Legendrian $Λ$ in $P\times\mathbb{R}$, such that the Chekanov-Eliashberg algebra of $\mathcal{T}(Λ,k)$ admits an augmentation, while $\mathcal{T}(Λ,k)$ has no exact embedded Lagrangian filling of vanishing Maslov class in the symplectization of $P\times \mathbb{R}$ for sufficiently large $k$.