Multisections with divides and Weinstein 4-manifolds

We show how to encode a Weinstein 4-manifold using a multisection diagram with divides, which is a sequence of cut systems on a surface, together with a separating collection of curves. We give two algorithms to construct a multisection diagram with divides of a given Weinstein manifold, one starting with a Kirby-Weinstein handle decomposition and one starting with a positive, allowable Lefschetz fibration (PALF). Through the connections with PALFs, we define a monodromy of a multisection and show how to symplectically carry out monodromy substitution on multisections with divides.