A User's Guide to Morse Foliated Open Books

Abstract: Morse foliated open books were introduced by the autors (along with abstract and embedded versions) as a tool for studying contact manifolds with boundary, and this article illustrates the advantages of the Morse perspective. We use this to extend the definition of \textit{right-veering} to foliated open books and we show that it plays a similar role in detecting overtwistedness as in other versions of open books.