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 condition of being standard. 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.arXiv:1610.06373v3 [math.GT]

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the original proof of the existence of relative trisections more explicit, in terms of handles. Furthermore, we extend this existence result to the case of 4manifolds with multiple boundary components, and show how trisected 4-manifolds with multiple boundary components glue together.

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We develop a technique for gluing relative trisection diagrams of 4-manifolds with nonempty connected boundary to obtain trisection diagrams for closed 4-manifolds.As an application, we describe a trisection of any closed 4-manifold which admits a Lefschetz fibration over S 2 equipped with a section of square −1, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed 4-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented S 2 -bundle over any closed surface and in particular we draw the corresponding diagrams for T 2 × S 2 and T 2 ×S 2 using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed 4-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.