# Properties of minimal charts and their applications VI: the graph $\Gamma_{m+1}$ in a chart $\Gamma$ of type $(m;2,3,2)$

@article{Nagase2020PropertiesOM, title={Properties of minimal charts and their applications VI: the graph \$\Gamma\_\{m+1\}\$ in a chart \$\Gamma\$ of type \$(m;2,3,2)\$}, author={Teruo Nagase and Akiko Shima}, journal={arXiv: Geometric Topology}, year={2020} }

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 $(m;2,3,2)$ if $w(\Gamma)=7$, $w(\Gamma_m\cap\Gamma_{m+1})=2$, $w(\Gamma_{m+1}\cap\Gamma_{m+2})=3$, 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 if there is a minimal chart $\Gamma$ of type $(m;2,3,2)$, then each of $\Gamma_{m+1}$ and $\Gamma_{m+2}$ contains one of three kinds of graphs. In the next…

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