论文标题
由桥梁三角诱导的立方图
Cubic graphs induced by bridge trisections
论文作者
论文摘要
4-Sphere中的每个嵌入式表面$ \ MATHCAL {K} $都允许一个桥梁三角，将$（s^4，\ Mathcal {k}）$的分解成三个简单的零件。在这种情况下，表面$ \ MATHCAL {K} $由嵌入式1个复合物确定，称为桥梁三角的$ \ textit {1-Skeleton} $。作为一个抽象图，1-Skeleton是一个立方图$γ$，它继承了天然的泰特尔着色，这是$γ$的边缘集的3色，使每个顶点都在所有三种颜色的边缘。在本文中，我们扭转了这种关联：我们证明，每个泰特颜色的立方图都是同构与与未打结的表面相对应的桥梁三角形的1骨骨骼。当表面不可定向时，我们表明，对于每个可能的正常欧拉号，都存在这样的嵌入。作为推论，可以通过交叉变化和内部雷迪德斯移动将打结表面的每个三平面图转换为三平面图，以进行未打结的表面。
Every embedded surface $\mathcal{K}$ in the 4-sphere admits a bridge trisection, a decomposition of $(S^4,\mathcal{K})$ into three simple pieces. In this case, the surface $\mathcal{K}$ is determined by an embedded 1-complex, called the $\textit{1-skeleton}$ of the bridge trisection. As an abstract graph, the 1-skeleton is a cubic graph $Γ$ that inherits a natural Tait coloring, a 3-coloring of the edge set of $Γ$ such that each vertex is incident to edges of all three colors. In this paper, we reverse this association: We prove that every Tait-colored cubic graph is isomorphic to the 1-skeleton of a bridge trisection corresponding to an unknotted surface. When the surface is nonorientable, we show that such an embedding exists for every possible normal Euler number. As a corollary, every tri-plane diagram for a knotted surface can be converted to a tri-plane diagram for an unknotted surface via crossing changes and interior Reidemeister moves.
