• 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…

## References

SHOWING 1-10 OF 18 REFERENCES
Properties of minimal charts and their applications VII: Charts of type (2,3,2)
• Mathematics
• 2020
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
Properties of minimal charts and their applications V: Charts of type (3,2,2)
• Mathematics
• 2019
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\$,
The structure of a minimal n-chart with two crossings I: Complementary domains of Γ1 ∪ Γn−1
• Mathematics
Journal of Knot Theory and Its Ramifications
• 2018
This is the first step of the two steps to enumerate the minimal charts with two crossings. For a label [Formula: see text] of a chart [Formula: see text] we denote by [Formula: see text] the union
Minimal charts of type
• Proc. Sch. Sci. TOKAI UNIV
• 2017
Properties of Minimal Charts and Their Applications IV: Loops
• Mathematics
• 2017
We investigate minimal charts with loops, a simple closed curve consisting of edges of label m containing exactly one white vertex. We shall show that there does not exist any loop in a minimal chart
Minimal charts of type (3,3)
• Mathematics
• 2016
Let \$\Gamma\$ be a chart. For each label \$m\$, we denote by \$\Gamma_m\$ the "subgraph" of \$\Gamma\$ consisting of all the edges of label \$m\$ and their vertices. Let \$\Gamma\$ be a minimal chart of type
THE CLOSURES OF SURFACE BRAIDS OBTAINED FROM MINIMAL n-CHARTS WITH FOUR WHITE VERTICES
• Mathematics
• 2013
In this paper, we investigate surface braids obtained from minimal charts with exactly four white vertices.
Properties of minimal charts and their applications II
• Mathematics
• 2009
We investigate minimal charts with loops, a simple closed curve consisting of edges of label \$m\$ containing exactly one white vertex. We shall show that there does not exist any loop in a minimal
Properties of minimal charts and their applications I
• Mathematics
• 2007
We study surface braids using the charts with minimal complexity. We introduce several terminology to describe minimal charts and investigate properties of minimal charts. We shall show that in a