{"title":"解结属 4 和 5 的非定向曲面","authors":"Mark Pencovitch","doi":"10.1016/j.laa.2024.08.014","DOIUrl":null,"url":null,"abstract":"<div><p>Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> with knot group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p><p>In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary <em>K</em> such that <span><math><mo>|</mo><mi>det</mi><mo></mo><mo>(</mo><mi>K</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary.</p><p>This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0024379524003410/pdfft?md5=b5b1d92c3f68749bd2133863f112514f&pid=1-s2.0-S0024379524003410-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Unknotting nonorientable surfaces of genus 4 and 5\",\"authors\":\"Mark Pencovitch\",\"doi\":\"10.1016/j.laa.2024.08.014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>4</mn></mrow></msup></math></span> with knot group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>.</p><p>In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary <em>K</em> such that <span><math><mo>|</mo><mi>det</mi><mo></mo><mo>(</mo><mi>K</mi><mo>)</mo><mo>|</mo><mo>=</mo><mn>1</mn></math></span>, and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary.</p><p>This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003410/pdfft?md5=b5b1d92c3f68749bd2133863f112514f&pid=1-s2.0-S0024379524003410-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003410\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003410","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
Unknotting nonorientable surfaces of genus 4 and 5
Expanding on work by Conway, Orson, and Powell, we study the isotopy classes rel. boundary of nonorientable, compact, locally flatly embedded surfaces in with knot group .
In particular we show that if two such surfaces have the same normal Euler number, the same fixed knot boundary K such that , and the same nonorientable genus 4 or 5, then they are ambiently isotopic rel. boundary.
This implies that closed, nonorientable, locally flatly embedded surfaces in the 4-sphere with knot group of nonorientable genus 4 and 5 are topologically unknotted. The proof relies on calculations, implemented in Sage, which imply that an obstruction to modified surgery is elementary. Furthermore we show that this method fails for nonorientable genus 6 and 7.
期刊介绍：
Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.
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