Bridge trisections of knotted surfaces in 4-manifolds
Bridge trisections of knotted surfaces in 4-manifolds
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DOI：
10.1073/pnas.1717171115
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
J. Meier;Alexander Zupan
中科院分区：
文献类型：
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作者：
J. Meier;Alexander Zupan
意义
低维拓扑学中的一个常见主题是将一个复杂的空间分解为简单的部分，并研究这些部分如何重新拼接在一起以恢复整个空间。例如，在标准三维空间\(R^3\)中一个纽结环的桥分解将该环切割成两组无纽结的弧。在四维空间中，有趣的纽结对象是曲面，并且在先前的工作中，作者融合了桥分解和三分剖分理论的思想来定义桥三分剖分，即标准四维空间\(R^4\)中纽结曲面的新型分解。在本文中，我们为更复杂的四维空间中的纽结曲面定义了广义桥三分剖分，为纽结曲面理论提供了一种不同的方法。我们证明，在一个\(4\)-流形中每一个光滑嵌入的曲面都可以通过同痕变换，相对于环境\(4\)-流形的一个给定三分剖分处于桥位置；也就是说，经过同痕变换后，该曲面与三分剖分的各个部分相交于平凡的圆盘或弧。这样一种分解，我们称之为广义桥三分剖分，它扩展了作者对于\(S^4\)中曲面的桥三分剖分的定义。利用这种构造，我们给出了\(4\)-流形中纽结曲面的一种称为阴影图的图示表示。我们还为这些结构提供了一种低复杂度的分类，并描述了几个例子，包括\(\mathbb{C}P^2\)内部复曲线的重要情形。利用这些例子，我们证明对于某些\(g\)的值，存在具有\((g,0)\)-三分剖分的奇异\(4\)-流形。最后我们简要描述了一个推测性的唯一性结果，它将为通过阴影图研究纽结曲面提供一个完整的图示计算方法。
Significance A common theme in low-dimensional topology is to split a complicated space into simple pieces and to study how these pieces can be glued back together to recover the total space. For example, a bridge splitting of a knotted loop in standard 3D space R3 cuts the loop into two collections of unknotted arcs. In dimension four, the interesting knotted objects are surfaces, and in previous work, the authors merged ideas from bridge splitting and trisection theories to define bridge trisections, novel decompositions of knotted surfaces in standard four-dimensional space R4. In this paper, we define generalized bridge trisections for knotted surfaces in more complicated four-dimensional spaces, offering a different approach to knotted surface theory. 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.