Properties of minimal charts and their applications X: charts of type $(5,2)$

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 $Γ$ be a chart, and we denote by $Γ_m$ the union of all the edges of label $m$. A chart $Γ$ is of type $(5,2)$ if there exists a label $m$ such that $w(Γ)=7$, $w(Γ_m\capΓ_{m+1})=5$, $w(Γ_{m+1}\capΓ_{m+2})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we investigate a minimal chart of type $(5,2)$.