John Etnyre: Embeddings of contact 3-manifolds

It is a well-known result of Hirsch that all 3-manifolds embed in\(R^5\). It was recently observed by Kasuya that if a contact 3-manifold embeds in the standard contact structure on \(R^5\) then its first Chern class must be zero. Getting a better understanding of such embeddings could be useful in the study of contact 5 manifolds (just as the study of transverse and Legendrian knots has been instrumental in the study of contact 3 manifolds). In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embedding in \(R^5\). Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss work with Yankı Lekili concerning embedding contact 3 manifolds in other contact 5 manifolds.