{"title":"Properties of minimal charts and their applications IX: charts of type (4,3)","authors":"Teruo Nagase , Akiko Shima","doi":"10.1016/j.indag.2023.01.009","DOIUrl":null,"url":null,"abstract":"<div><p>Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let <span><math><mi>Γ</mi></math></span> be a chart, and we denote by <span><math><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> the union of all the edges of label <span><math><mi>m</mi></math></span>. A chart <span><math><mi>Γ</mi></math></span> is of type <span><math><mrow><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></math></span> if there exists a label <span><math><mi>m</mi></math></span> such that <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>Γ</mi><mo>)</mo></mrow><mo>=</mo><mn>7</mn></mrow></math></span>, <span><math><mrow><mi>w</mi><mrow><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∩</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>, <span><math><mrow><mi>w</mi><mrow><mo>(</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>Γ</mi></mrow><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>3</mn></mrow></math></span> where <span><math><mrow><mi>w</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the number of white vertices in <span><math><mi>G</mi></math></span>. In this paper, we prove that there is no minimal chart of type <span><math><mrow><mo>(</mo><mn>4</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0019357723000113","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let be a chart, and we denote by the union of all the edges of label . A chart is of type if there exists a label such that , , where is the number of white vertices in . In this paper, we prove that there is no minimal chart of type .
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.