{"title":"Calculation of Nielsen periodic numbers on infra-solvmanifolds","authors":"","doi":"10.1016/j.topol.2024.109073","DOIUrl":"10.1016/j.topol.2024.109073","url":null,"abstract":"<div><div>Recently, a formula for computing the Nielsen periodic numbers <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> and <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of self maps <em>f</em> on infra-nilmanifolds and infra-solvmanifolds of type (R) was found. In this paper, we extend this formula to the case of general infra-solvmanifolds. We show that infra-solvmanifolds are essentially reducible to the GCD and essentially toral, and determine conditions under which <span><math><mi>N</mi><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>(</mo><msup><mrow><mi>f</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. We show that the prime Nielsen-Jiang periodic number <span><math><mi>N</mi><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>f</mi><mo>)</mo></math></span> of a self map <em>f</em> on an infra-solvmanifold <em>M</em> can be calculated by Nielsen numbers of lifts of suitable iterates of <em>f</em> to an <span><math><mi>NR</mi></math></span>-solvmanifold that finitely covers <em>M</em>.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Hereditarily decomposable continua have non-block points","authors":"","doi":"10.1016/j.topol.2024.109072","DOIUrl":"10.1016/j.topol.2024.109072","url":null,"abstract":"<div><div>In this note we expand upon our results from <span><span>[1]</span></span> to show that every nondegenerate hereditarily decomposable Hausdorff continuum has two or more non-block points, i.e. points whose complements contain a continuum-connected dense subset. The celebrated non-cut point existence theorem states that all nondegenerate Hausdorff continua have two or more non-cut points, and the corresponding result for non-block points is known to hold for metrizable continua. It is also known that there are consistent examples of Hausdorff continua with no non-block points, but that non-block point existence holds for Hausdorff continua that are either aposyndetic, irreducible, or separable.</div></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Jones polynomial for a torus knot with twists","authors":"","doi":"10.1016/j.topol.2024.109069","DOIUrl":"10.1016/j.topol.2024.109069","url":null,"abstract":"<div><p>We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form <span><math><mi>T</mi><mo>(</mo><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>,</mo><mo>(</mo><mn>2</mn><mo>,</mo><mi>s</mi><mo>)</mo><mo>)</mo></math></span> where <em>p</em> and <em>q</em> are coprime and <em>s</em> is nonzero. When <span><math><mi>s</mi><mo>=</mo><mn>2</mn><mi>n</mi></math></span>, these links are the twisted torus knots <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>;</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>. We show that for <span><math><mi>T</mi><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>;</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>, the Jones polynomial is trivial if and only if the knot is trivial.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"More on Whitney levels of some decomposable continua","authors":"","doi":"10.1016/j.topol.2024.109068","DOIUrl":"10.1016/j.topol.2024.109068","url":null,"abstract":"<div><p>In this paper, we show that there exists a non-<em>D</em>-continuum such that each positive Whitney level of the hyperspace of subcontinua of the continuum is both <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> and Wilder. We show that the property of being continuum-wise Wilder is not a Whitney property, while it is a Whitney reversible property. Furthermore, we introduce the new class of continua: closed set-wise Wilder continua. This class is larger than the class of continuum chainable continua and smaller than the class of continuum-wise Wilder continua. In addition to the above results, we show that the Cartesian product of two closed set-wise Wilder continua is close set-wise Wilder.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142242463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"One-point connectifications of regular spaces","authors":"","doi":"10.1016/j.topol.2024.109060","DOIUrl":"10.1016/j.topol.2024.109060","url":null,"abstract":"<div><p>It is well known that, a locally compact Hausdorff space has a Hausdorff one-point compactification (known as the <em>Alexandroff compactification</em>) if and only if it is non-compact. There is also, an old question of Alexandroff of characterizing spaces which have a one-point connectification. Here, we study one-point connectifications in the realm of regular spaces and prove that a locally connected space has a regular one-point connectification if and only if the space has no regular-closed component. This, also gives an answer to the conjecture raised by M. R. Koushesh. Then, we consider the set of all one-point connectifications of a locally connected regular space and show that, this set (naturally partially ordered) is a compact conditionally complete lattice. Further, we extend our theorem for locally connected regular spaces with a topological property <span><math><mi>P</mi></math></span> and give conditions on <span><math><mi>P</mi></math></span> which guarantee the space to have a regular one-point connectification with <span><math><mi>P</mi></math></span>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Macías topology on integral domains","authors":"","doi":"10.1016/j.topol.2024.109070","DOIUrl":"10.1016/j.topol.2024.109070","url":null,"abstract":"<div><p>In this manuscript a recent topology on the positive integers generated by the collection of <span><math><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo></math></span> where <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>:</mo><mo>=</mo><mo>{</mo><mi>m</mi><mo>:</mo><mi>gcd</mi><mo></mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>}</mo></math></span> is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A family of slice-torus invariants from the divisibility of Lee classes","authors":"","doi":"10.1016/j.topol.2024.109059","DOIUrl":"10.1016/j.topol.2024.109059","url":null,"abstract":"<div><p>We give a family of slice-torus invariants <span><math><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span>, each defined from the <em>c</em>-divisibility of the reduced Lee class in a variant of reduced Khovanov homology, parameterized by prime elements <em>c</em> in any principal ideal domain <em>R</em>. For the special case <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>F</mi><mo>[</mo><mi>H</mi><mo>]</mo><mo>,</mo><mi>H</mi><mo>)</mo></math></span> where <em>F</em> is any field, we prove that <span><math><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span> coincides with the Rasmussen invariant <span><math><msup><mrow><mi>s</mi></mrow><mrow><mi>F</mi></mrow></msup></math></span> over <em>F</em>. Compared with the unreduced invariants <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> defined by the first author in a previous paper, we prove that <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>=</mo><msub><mrow><mover><mrow><mi>s</mi><mi>s</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>c</mi></mrow></msub></math></span> for <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>F</mi><mo>[</mo><mi>H</mi><mo>]</mo><mo>,</mo><mi>H</mi><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>Z</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span>. However for <span><math><mo>(</mo><mi>R</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>Z</mi><mo>,</mo><mn>3</mn><mo>)</mo></math></span>, computational results show that <span><math><mi>s</mi><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> is not slice-torus, which implies that it is linearly independent from the reduced invariants, and particularly from the Rasmussen invariants.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142204846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Small difference between tunnel numbers of cable knots and their companions","authors":"","doi":"10.1016/j.topol.2024.109058","DOIUrl":"10.1016/j.topol.2024.109058","url":null,"abstract":"<div><p>We prove that for any nontrivial knot <span><math><mi>K</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> and a <span><math><mi>p</mi><mo>/</mo><mi>q</mi></math></span>-cable knot <span><math><msup><mrow><mi>K</mi></mrow><mrow><mo>⋆</mo></mrow></msup></math></span> of <em>K</em>, the tunnel number <span><math><mi>t</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>=</mo><mi>t</mi><mo>(</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>⋆</mo></mrow></msup><mo>)</mo></math></span> if and only if <em>K</em> is <span><math><mi>p</mi><mo>/</mo><mi>q</mi></math></span>-primitive. This result solves a problem mentioned in <span><span>[8]</span></span>.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142173848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Max(dL) revisited","authors":"","doi":"10.1016/j.topol.2024.109057","DOIUrl":"10.1016/j.topol.2024.109057","url":null,"abstract":"<div><p>This article studies different topological properties of the space of maximal <em>d</em>-elements of an <em>M</em>-frame with a unit. We characterize when the space <span><math><mi>M</mi><mi>a</mi><mi>x</mi><mo>(</mo><mi>d</mi><mi>L</mi><mo>)</mo></math></span> is Hausdorff, answering the question posed in <span><span>[2]</span></span>. We also characterize other topological properties of <span><math><mi>M</mi><mi>a</mi><mi>x</mi><mo>(</mo><mi>d</mi><mi>L</mi><mo>)</mo></math></span>, namely zero-dimensional, discrete, and clopen <em>π</em>-base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of <em>d</em>-semiprime frames.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142164475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The local-to-global principle via topological properties of the tensor triangular support","authors":"","doi":"10.1016/j.topol.2024.109056","DOIUrl":"10.1016/j.topol.2024.109056","url":null,"abstract":"<div><p>Following the theory of tensor triangular support introduced by Sanders, which generalizes the Balmer-Favi support, we prove the local version of the result of Zou that the Balmer spectrum being Hochster weakly scattered implies the local-to-global principle.</p><p>That is, given an object <em>t</em> of a tensor triangulated category <span><math><mi>T</mi></math></span> we show that if the tensor triangular support <span><math><mtext>Supp</mtext><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is a weakly scattered subset with respect to the inverse topology of the Balmer spectrum <span><math><mtext>Spc</mtext><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span>, then the local-to-global principle holds for <em>t</em>.</p><p>As immediate consequences, we have the analogue adaptations of the well-known statements that the Balmer spectrum being noetherian or Hausdorff scattered implies the local-to-global principle.</p><p>We conclude with an application of the last result to the examination of the support of injective superdecomposable modules in the derived category of an absolutely flat ring which is not semi-artinian.</p></div>","PeriodicalId":51201,"journal":{"name":"Topology and its Applications","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142122765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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