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

39 pages, 30 figures