• Corpus ID: 214667334

# 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}
}```
• Published 25 March 2020
• Mathematics
• arXiv: Geometric Topology
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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Abstract Let Γ be a chart, and we denote by Γ m the union of all the edges of label m. A chart Γ is of type ( 2 , 3 , 2 ) if there exists a label m such that w ( Γ ) = 7 , w ( Γ m ∩ Γ m + 1 ) = 2 , w
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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\$,
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