Properties of minimal charts and their applications IX: charts of type (4,3)

Abstract

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional 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 $\Gamma$ be a chart, and we denote by ${\Gamma }_m$ the union of all the edges of label $m$. A chart $\Gamma$ is of type $(4,3)$ if there exists a label $m$ such that $w\left(\Gamma \right)=7$, $w({\Gamma }_m\cap {\Gamma }_{m+1})=4$, $w({\Gamma }_{m+1}\cap {\Gamma }_{m+2})=3$ where $w\left(G\right)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(4,3)$.