In this talk we will explain how to use the diagrammatic calculus of Demazure weaves to construct and manipulate relative CY3 categories associated to certain Lefschetz fibrations. A key idea is the introduction and study of categorical semitwists and the symplectic geometric realization of simple heart tilts. In a nutshell, we will explain developments in and around the interaction between contact and symplectic geometry and the representation theory surrounding cluster algebras.