Diagrams for Relative Trisections

We establish a correspondence between trisections of smooth, compact, oriented $4$–manifolds with connected boundary and diagrams describing these trisected $4$–manifolds. Such a diagram comes in the form of a compact, oriented surface with boundary together with three tuples of simple closed curves, with possibly fewer curves than the genus of the surface, satisfying a pairwise standardness condition. This should be thought of as the $4$–dimensional analog of a sutured Heegaard diagram for a sutured $3$–manifold. We also give many foundational examples.