Diagrams for Relative Trisections

Abstract

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.

The previous version had mistakes in the proof of the algorithm to find the monodromy of the induced open book decomposition using the relative trisection diagram. This version improves the statement of the algorithm and corrects the proof