A NOTE ON OPEN BOOK EMBEDDINGS OF -MANIFOLDS IN

In this note, we show that given a closed connected oriented $3$ -manifold M, there exists a knot K in M such that the manifold $M'$ obtained from M by performing an integer surgery admits an open book decomposition which embeds into the trivial open book of the $5$ -sphere $S^5.$

On Open Book Embedding of Contact Manifolds in the Standard Contact Sphere

We prove some open book embedding results in the contact category with a constructive approach. As a consequence, we give an alternative proof of a theorem of Etnyre and Lekili that produces a large class of contact 3-manifolds admitting contact open book embeddings in the standard contact 5-sphere. We also show that all the Ustilovsky $(4m+1)$-spheres contact open book embed in the standard contact $(4m+3)$-sphere.