Bridge trisections of knotted surfaces in 4-manifolds

Abstract

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 generalized bridge trisection, extends the authors’ definition of bridge trisections for surfaces in S4. Using this construction, we give diagrammatic representations called 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 ℂℙ2. 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.