Properties of minimal charts and their applications VIII: charts of type $(7)$

TL;DR

This paper proves that minimal charts of type (7), characterized by specific label and vertex intersection properties, do not exist, advancing understanding of chart classifications in topological studies.

Contribution

It establishes the non-existence of minimal charts of type (7), providing a key classification result in the theory of charts.

Findings

No minimal chart of type (7) exists.

Clarifies the structure and limitations of chart types.

Contributes to the classification of minimal charts.

Abstract

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 $(7)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=7$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(7)$.