Properties of minimal charts and their applications V: charts of type $(3,2,2)$

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 $(3,2,2)$ if there exists a label $m$ such that $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=3$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=2$, and $w(\Gamma_{m+2}\cap\Gamma_{m+3})=2$ where $w(G)$ is the number of white vertices in $G$. In this paper, we prove that there is no minimal chart of type $(3,2,2)$.
Comment: 27 pages, 23 figures. arXiv admin note: text overlap with arXiv:1609.08257, arXiv:1603.04639