论文标题
最小图表的属性及其应用IX：类型$（4,3）$的图表
Properties of minimal charts and their applications IX: charts of type $(4,3)$
论文作者
论文摘要
图表是磁盘中的标记图。可以使用图表来描述任何简单的表面编织（2-二甲状腺编织）。此外，图表表示嵌入4空间的定向闭合表面。在本文中，我们使用图表研究了4空间中的嵌入式表面。令$γ$为图表，我们用$γ_m$表示标签$ m $的所有边缘的结合。如果存在标签$ m $，则图表$γ$是类型$（4,3）$的类型（γ_M\capγ_{m+1}）= 4 $，$ w（γ_{m+1} \capγ_{m+2}）= 3 $ w $ w（g）$ withe $ with，在本文中，我们证明没有最小的$（4,3）$的图表。
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensonal braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $Γ$ be a chart, and we denote by $Γ_m$ the union of all the edges of label $m$. A chart $Γ$ is of type $(4,3)$ if there exists a label $m$ such that $w(Γ)=7$, $w(Γ_m\capΓ_{m+1})=4$, $w(Γ_{m+1}\capΓ_{m+2})=3$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(4,3)$.
