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

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 $(7)$.

40 pages, 30 figures. arXiv admin note: text overlap with arXiv:2003.12188, arXiv:2003.11909, arXiv:1902.00007, arXiv:1603.04639, arXiv:1609.08257