# On Legendrian Embbeddings into Open Book Decompositions

## TL;DR

This paper proves that Legendrian submanifolds in certain contact manifolds supported by Weinstein open books can be isotoped to avoid the pages, revealing new flexibility properties of Legendrian embeddings.

## Contribution

It establishes a method to Legendrian isotope submanifolds to be disjoint from open book pages in contact manifolds with Weinstein pages, under isotopic contact structures.

## Key findings

- Legendrian submanifolds can be isotoped to avoid open book pages
- Existence of a contactomorphism making the submanifold disjoint from a page
- Applicable to contact structures supported by Weinstein open books

## Abstract

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,\xi)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $\xi'$ on $M$, isotopic to $\xi$ and supported by $\mathcal{OB}$, and a contactomorphism $f:(M,\xi) \to (M,\xi')$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.