论文标题
填充编织的链接，带有三角形的表面
Filling braided links with trisected surfaces
论文作者
论文摘要
我们介绍了紧凑的四个manifold中整齐的嵌入式表面的桥梁进行概念，在封闭的四个manifolds中的封闭表面上概括了先前与亚历山大·祖帕（Alexander Zupan）的先前工作。我们的主要结果指出，在紧凑的四个manifold $ x $中，任何整齐的嵌入式表面$ \ MATHCAL {f} $都可以同位素相对于任何Trisection $ \ Mathbb {t} $ of $ x $。 $ \ MATHCAL {F} $的桥梁转移引起了链接$ \ partial \ Mathcal {f} $的链接，相对于$ \ partial x $的开簿分解，由$ \ mathbb {t} $引起的$ \ partial x $的分解，我们表明了$ \ mathcal的桥梁trisection to to to n braid to to f} $。我们在可能断开$ \ partial x $的一般环境中工作，我们描述了如何使用阴影图来编码桥梁的表面图。我们使用阴影图来显示如何沿其边界的部分粘合桥梁的表面，并说明如何从阴影图中恢复边界链接的编织数据。在整个过程中，给出了许多例子和插图。我们给出了一组我们猜想的动作，足以将与给定表面相对应的任何两个阴影图相关联。我们将额外的注意力集中在$ b^4 $中的表面设置，在这里我们为存在桥梁三触角的存在独立证明，并使用三平面图开发了第二种图形方法。我们根据色带表面的复杂性参数来表征桥面的桥梁三触角。详细介绍了$ b^4 $中的桥梁三触角和频段演示的过程，并提供了许多示例。
We introduce the concept of a bridge trisection of a neatly embedded surface in a compact four-manifold, generalizing previous work with Alexander Zupan in the setting of closed surfaces in closed four-manifolds. Our main result states that any neatly embedded surface $\mathcal{F}$ in a compact four-manifold $X$ can be isotoped to lie in bridge trisected position with respect to any trisection $\mathbb{T}$ of $X$. A bridge trisection of $\mathcal{F}$ induces a braiding of the link $\partial\mathcal{F}$ with respect to the open-book decomposition of $\partial X$ induced by $\mathbb{T}$, and we show that the bridge trisection of $\mathcal{F}$ can be assumed to induce any such braiding. We work in the general setting in which $\partial X$ may be disconnected, and we describe how to encode bridge trisected surface diagrammatically using shadow diagrams. We use shadow diagrams to show how bridge trisected surfaces can be glued along portions of their boundary, and we explain how the data of the braiding of the boundary link can be recovered from a shadow diagram. Throughout, numerous examples and illustrations are given. We give a set of moves that we conjecture suffice to relate any two shadow diagrams corresponding to a given surface. We devote extra attention to the setting of surfaces in $B^4$, where we give an independent proof of the existence of bridge trisections and develop a second diagrammatic approach using tri-plane diagrams. We characterize bridge trisections of ribbon surfaces in terms of their complexity parameters. The process of passing between bridge trisections and band presentations for surfaces in $B^4$ is addressed in detail and presented with many examples.
