Bridge trisections of knotted surfaces in 4--manifolds

We prove that every smoothly embedded surface in a 4–manifold can be isotoped to be in bridge position with respect to a given trisection of the ambient 4–manifold; that is, after isotopy, the surface meets components of the trisection in trivial disks or arcs. Such a decomposition, which we call a \emph{generalized bridge trisection}, extends the authors’ definition of bridge trisections for surfaces in $S^4$. Using this new construction, we give diagrammatic representations called \emph{shadow diagrams} for knotted surfaces in 4–manifolds. We also provide a low-complexity classification for these structures and describe several examples, including the important case of complex curves inside ${\mathbb{C}\mathbb{P}}^2$. Using these examples, we prove that there exist exotic 4–manifolds with $(g,0)$–trisections for certain values of $g$. We conclude by sketching a conjectural uniqueness result that would provide a complete diagrammatic calculus for studying knotted surfaces through their shadow diagrams.