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 S 4 . Using this new 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 CP 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.

Abstract. We show that the members of a large class of unbalanced four-manifold trisections are standard, and we present a family of trisections that is likely to include non-standard trisections of the four-sphere. As an application, we prove a stable version of the Generalized Property R Conjecture for c-component links with tunnel number at most c.

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of fourmanifolds and extends the classical concept of bridge splittings of links in the threesphere to four dimensions. We prove that every knotted surface in the four-sphere admits a bridge trisection (a decomposition into three simple pieces) and that any two bridge trisections for a fixed surface are related by a sequence of stabilizations and destabilizations. We also introduce a corresponding diagrammatic representation of knotted surfaces and describe a set of moves that suffice to pass between two diagrams for the same surface. Using these decompositions, we define a new complexity measure: the bridge number of a knotted surface. In addition, we classify bridge trisections with low complexity, we relate bridge trisections to the fundamental groups of knotted surface complements, and we prove that there exist knotted surfaces with arbitrarily large bridge number. arXiv:1507.08370v1 [math.GT]

We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in [Formula: see text] and [Formula: see text]. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call efficient. We show that any curve in [Formula: see text] or [Formula: see text] admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of complex surfaces that admit efficient trisections. Among these are hypersurfaces in [Formula: see text], the elliptic surfaces [Formula: see text], the Horikawa surfaces [Formula: see text], and complete intersections of hypersurfaces in [Formula: see text]. As a corollary, we observe that, in many cases, manifolds that are homeomorphic but not diffeomorphic have the same trisection genus, which is consistent with the conjecture that trisection genus is additive under connected sum. We give many trisection diagrams to illustrate our examples.

We show that the only closed 4-manifolds admitting genus two trisections are S 2 ×S 2 and connected sums of S 1 ×S 3 , CP 2 , and CP 2 with two summands.Moreover, each of these manifolds admits a unique genus two trisection up to diffeomorphism. The proof relies heavily on the combinatorics of genus two Heegaard diagrams of S 3 . As a corollary, we classify two-component links contained in a genus two Heegaard surface for S 3 with a surface-sloped cosmetic Dehn surgery.