Department of Mathematics, University of Georgia, Athens, GA 30602;
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588.
Proc Natl Acad Sci U S A. 2018 Oct 23;115(43):10880-10886. doi: 10.1073/pnas.1717171115.
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 [Formula: see text] 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 [Formula: see text] Using these examples, we prove that there exist exotic 4-manifolds with [Formula: see text]-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.
我们证明了，在一个 4 维流形中，每一个光滑嵌入的曲面都可以通过对环境 4 维流形的给定三分来进行同胚，使得在同胚后，曲面与三分的分量相交于平凡圆盘或弧。这种分解，我们称之为广义桥三分，扩展了作者在 [Formula: see text] 中对曲面的桥三分的定义。利用这种构造，我们给出了 4 维流形中打结曲面的称为阴影图的图示表示。我们还对这些结构进行了低复杂度分类，并描述了几个例子，包括 [Formula: see text] 内部复杂曲线的重要情况。利用这些例子，我们证明了在某些 g 值下存在具有 [Formula: see text]-三分的外来 4 维流形。最后，我们通过概述一个推测的唯一性结果来结束，该结果将为通过阴影图研究打结曲面提供一个完整的图示演算。
