Properties of Minimal Charts and their Applications VI: The Graph $\boldsymbol{\Gamma_{m+1}}$ in a Chart $\boldsymbol{\Gamma}$ of Type $(m;2,3,2)$

J. Math. Sci. Univ. Tokyo
Vol. 27 (2020), No. 1, Page 109-156.

Nagase, Teruo; Shima, Akiko
Properties of Minimal Charts and their Applications VI: The Graph $\boldsymbol{\Gamma_{m+1}}$ in a Chart $\boldsymbol{\Gamma}$ of Type $(m;2,3,2)$
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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 $(m;2,3,2)$ 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 paper, we shall prove that there is no minimal chart of type $(m;2,3,2)$.

Keywords: Surface link, chart, white vertex.

Mathematics Subject Classification (2010): Primary 57Q45; Secondary 57Q35.
Received: 2020-04-07