{"title":"Some integer values in the spectra of burnt pancake graphs","authors":"Saúl A. Blanco , Charles Buehrle","doi":"10.1016/j.laa.2024.09.003","DOIUrl":"10.1016/j.laa.2024.09.003","url":null,"abstract":"<div><p>The burnt pancake graph, denoted by <span><math><msub><mrow><mi>BP</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, is formed by connecting signed permutations via prefix reversals. Here, we discuss some spectral properties of <span><math><msub><mrow><mi>BP</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. More precisely, we prove that the adjacency spectrum of <span><math><msub><mrow><mi>BP</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> contains all integer values in the set <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo><mo>∖</mo><mo>{</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow><mo>}</mo></math></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 163-172"},"PeriodicalIF":1.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142171894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The n-th production matrix of a Riordan array","authors":"Hong-Zhang Ai, Xun-Tuan Su","doi":"10.1016/j.laa.2024.08.022","DOIUrl":"10.1016/j.laa.2024.08.022","url":null,"abstract":"<div><p>The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the <em>n</em>-th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the <em>n</em>-th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every <em>n</em>-th production matrix can be factorized into a product of <em>n</em> matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the <em>n</em>-th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the <em>n</em>-th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 63-77"},"PeriodicalIF":1.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On stabilizing index and cyclic index of certain amalgamated uniform hypergraphs","authors":"Cheng Yeaw Ku , Kok Bin Wong","doi":"10.1016/j.laa.2024.08.020","DOIUrl":"10.1016/j.laa.2024.08.020","url":null,"abstract":"<div><p>Let <em>G</em> be a connected uniform hypergraph and <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the adjacency tensor of <em>G</em>. The largest absolute value of the eigenvalues of <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is called the spectral radius of <em>G</em>. The number of eigenvectors of <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> associated with the spectral radius is called the stabilizing index of <em>G</em>. The number of eigenvalues of <span><math><mi>A</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> with modulus equal to the spectral radius is called the cyclic index of <em>G</em>. In this paper, we consider a class of amalgamated uniform hypergraphs and compute its stabilizing index and cyclic index.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 109-136"},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142169244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The eigenvalue multiplicity of line graphs","authors":"Sarula Chang , Jianxi Li , Yirong Zheng","doi":"10.1016/j.laa.2024.08.021","DOIUrl":"10.1016/j.laa.2024.08.021","url":null,"abstract":"<div><p>Let <span><math><mi>m</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>, <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and <span><math><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the multiplicity of an eigenvalue <em>λ</em>, the cyclomatic number and the number of pendant vertices of a connected graph <em>G</em>, respectively. Yang et al. (2023) <span><span>[10]</span></span> proved that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any tree <em>T</em>, and characterized all trees <em>T</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mi>p</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is the line graph of <em>T</em>. In this paper, we extend their result from a tree <em>T</em> to any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and prove that <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> for any graph <span><math><mi>G</mi><mo>≠</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Moreover, all graphs <em>G</em> with <span><math><mi>m</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>,</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>2</mn><mi>c</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span> are completely characterized.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 47-62"},"PeriodicalIF":1.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142163933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Majorization in some symplectic weak supermajorizations","authors":"Shaowu Huang , Hemant K. Mishra","doi":"10.1016/j.laa.2024.08.019","DOIUrl":"10.1016/j.laa.2024.08.019","url":null,"abstract":"<div><p>Symplectic eigenvalues are known to satisfy analogs of several classic eigenvalue inequalities. Of these is a set of weak supermajorization relations concerning symplectic eigenvalues that are weaker analogs of some majorization relations corresponding to eigenvalues. The aim of this letter is to establish necessary and sufficient conditions for the saturation of the symplectic weak supermajorization relations by majorization.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 1-10"},"PeriodicalIF":1.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Normal approximations of commuting square-summable matrix families","authors":"Alexandru Chirvasitu","doi":"10.1016/j.laa.2024.08.017","DOIUrl":"10.1016/j.laa.2024.08.017","url":null,"abstract":"<div><p>For any square-summable commuting family <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></math></span> of complex <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices there is a normal commuting family <span><math><msub><mrow><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>i</mi></mrow></msub></math></span> no farther from it, in squared normalized <span><math><msup><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> distance, than the diameter of the numerical range of <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>i</mi></mrow></msub><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. Specializing in one direction (limiting case of the inequality for finite <em>I</em>) this recovers a result of M. Fraas: if <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msubsup><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a multiple of the identity for commuting <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>C</mi><mo>)</mo></math></span> then the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> are normal; specializing in another (singleton <em>I</em>) retrieves the well-known fact that close-to-isometric matrices are close to isometries.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 11-19"},"PeriodicalIF":1.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Contractivity of Möbius functions of operators","authors":"Thomas Ransford , Dashdondog Tsedenbayar","doi":"10.1016/j.laa.2024.08.018","DOIUrl":"10.1016/j.laa.2024.08.018","url":null,"abstract":"<div><p>Let <em>T</em> be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers <span><math><mi>λ</mi><mo>,</mo><mi>μ</mi></math></span> for which <span><math><mo>(</mo><mi>I</mi><mo>+</mo><mi>λ</mi><mi>T</mi><mo>)</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>μ</mi><mi>T</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>.</p><p>When <span><math><mi>T</mi><mo>=</mo><mi>V</mi></math></span>, the Volterra operator on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those <span><math><mi>λ</mi><mo>,</mo><mi>μ</mi></math></span> for which <span><math><mo>(</mo><mi>I</mi><mo>+</mo><mi>λ</mi><mi>V</mi><mo>)</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>μ</mi><mi>V</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> is a contraction. Taking <span><math><mi>T</mi><mo>=</mo><msup><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, we further deduce that <span><math><mo>(</mo><mi>I</mi><mo>+</mo><mi>λ</mi><msup><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><msup><mrow><mo>(</mo><mi>I</mi><mo>+</mo><mi>μ</mi><msup><mrow><mi>V</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> is never a contraction if <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>λ</mi><mo>≠</mo><mi>μ</mi></math></span>.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 20-26"},"PeriodicalIF":1.0,"publicationDate":"2024-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Gi-Sang Cheon , Bumtle Kang , Suh-Ryung Kim , Homoon Ryu
{"title":"Matrix periods and competition periods of Boolean Toeplitz matrices II","authors":"Gi-Sang Cheon , Bumtle Kang , Suh-Ryung Kim , Homoon Ryu","doi":"10.1016/j.laa.2024.08.016","DOIUrl":"10.1016/j.laa.2024.08.016","url":null,"abstract":"<div><p>This paper is a follow-up to the paper of Cheon et al. (2023) <span><span>[2]</span></span>. Given subsets <em>S</em> and <em>T</em> of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>}</mo></math></span>, an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> Toeplitz matrix <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>S</mi><mo>;</mo><mi>T</mi><mo>〉</mo></math></span> is defined to have 1 as the <span><math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math></span>-entry if and only if <span><math><mi>j</mi><mo>−</mo><mi>i</mi><mo>∈</mo><mi>S</mi></math></span> or <span><math><mi>i</mi><mo>−</mo><mi>j</mi><mo>∈</mo><mi>T</mi></math></span>. In the previous paper, we have shown that the matrix period and the competition period of Toeplitz matrices <span><math><mi>A</mi><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>S</mi><mo>;</mo><mi>T</mi><mo>〉</mo></math></span> satisfying the condition (⋆) <span><math><mi>max</mi><mo></mo><mi>S</mi><mo>+</mo><mi>min</mi><mo></mo><mi>T</mi><mo>≤</mo><mi>n</mi></math></span> and <span><math><mi>min</mi><mo></mo><mi>S</mi><mo>+</mo><mi>max</mi><mo></mo><mi>T</mi><mo>≤</mo><mi>n</mi></math></span> are <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>/</mo><mi>d</mi></math></span> and 1, respectively, where <span><math><msup><mrow><mi>d</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>=</mo><mi>gcd</mi><mo></mo><mo>(</mo><mi>s</mi><mo>+</mo><mi>t</mi><mo>|</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>T</mi><mo>)</mo></math></span> and <span><math><mi>d</mi><mo>=</mo><mi>gcd</mi><mo></mo><mo>(</mo><mi>d</mi><mo>,</mo><mi>min</mi><mo></mo><mi>S</mi><mo>)</mo></math></span>. In this paper, we claim that even if (⋆) is relaxed to the existence of elements <span><math><mi>s</mi><mo>∈</mo><mi>S</mi></math></span> and <span><math><mi>t</mi><mo>∈</mo><mi>T</mi></math></span> satisfying <span><math><mi>s</mi><mo>+</mo><mi>t</mi><mo>≤</mo><mi>n</mi></math></span> and <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, the same result holds. There are infinitely many Toeplitz matrices that do not satisfy (⋆) but the relaxed condition. For example, for any positive integers <span><math><mi>k</mi><mo>,</mo><mi>n</mi></math></span> with <span><math><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>≤</mo><mi>n</mi></math></span>, it is easy to see that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>〈</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>;</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>〉</mo></math></span> does not satisfy (⋆) but satisfies the relaxed condition. Furthermore, we show that the limit of the matrix sequence <span><math><msubsup><mrow><mo>{</mo><","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"703 ","pages":"Pages 27-46"},"PeriodicalIF":1.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142148823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Minimizing the Laplacian-energy-like of graphs","authors":"Gao-Xuan Luo, Shi-Cai Gong , Jing Tian","doi":"10.1016/j.laa.2024.08.015","DOIUrl":"10.1016/j.laa.2024.08.015","url":null,"abstract":"<div><p>Let <em>G</em> be a connected simple graph with order <em>n</em> and Laplacian matrix <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The Laplacian-energy-like of <em>G</em> is defined as<span><span><span><math><mi>L</mi><mi>E</mi><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><msqrt><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msqrt><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>. In this paper, with the aid of Ferrers diagrams of threshold graphs, we provide an algebraic combinatorial approach to determine the graphs with minimal Laplacian-energy-like among all connected graphs having <em>n</em> vertices and <em>m</em> edges, showing that the extremal graph is a special threshold graph, named as the quasi-complete graph.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"702 ","pages":"Pages 179-194"},"PeriodicalIF":1.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unknotting nonorientable surfaces of genus 4 and 5","authors":"Mark Pencovitch","doi":"10.1016/j.laa.2024.08.014","DOIUrl":"10.1016/j.laa.2024.08.014","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":"702 ","pages":"Pages 195-217"},"PeriodicalIF":1.0,"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":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142097783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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