Critical point counts in knot cobordisms: abelian and metacyclic invariants

Transactions of the American Mathematical Society, Series B Pub Date : 2021-06-30 DOI:10.1090/btran/139

C. Livingston

{"title":"Critical point counts in knot cobordisms: abelian and metacyclic invariants","authors":"C. Livingston","doi":"10.1090/btran/139","DOIUrl":null,"url":null,"abstract":"<p>For a pair of knots <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">K_1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 0\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">K_0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, we consider the set of four-tuples of integers <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis g comma c 0 comma c 1 comma c 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>g</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>c</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>c</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:mo>,</mml:mo>\n <mml:msub>\n <mml:mi>c</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msub>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">(g, c_0,c_1, c_2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> for which there is a cobordism from <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 1\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mn>1</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">K_1</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> to <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K 0\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>K</mml:mi>\n <mml:mn>0</mml:mn>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">K_0</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> of genus <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"g\">\n <mml:semantics>\n <mml:mi>g</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">g</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> having <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c Subscript i\">\n <mml:semantics>\n <mml:msub>\n <mml:mi>c</mml:mi>\n <mml:mi>i</mml:mi>\n </mml:msub>\n <mml:annotation encoding=\"application/x-tex\">c_i</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> critical points of each index <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"i\">\n <mml:semantics>\n <mml:mi>i</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">i</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. We describe basic properties that such sets must satisfy and then build homological obstructions to membership in the set. These obstructions are determined by knot invariants arising from cyclic and metacyclic covering spaces.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"37 36","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/139","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 1

Abstract

For a pair of knots K 1 K_1 and K 0 K_0 , we consider the set of four-tuples of integers ( g , c 0 , c 1 , c 2 ) (g, c_0,c_1, c_2) for which there is a cobordism from K 1 K_1 to K 0 K_0 of genus g g having c i c_i critical points of each index i i . We describe basic properties that such sets must satisfy and then build homological obstructions to membership in the set. These obstructions are determined by knot invariants arising from cyclic and metacyclic covering spaces.

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结协中的临界点计数:阿贝尔和亚循环不变量

对于一对结点k1k_1和k0k_0，我们考虑整数(g, c0, c1, c2) (g, c_0,c_1，c_2)它有一个从k1k_1到k0k_0的g g格的协配，每个指标i i都有c_i个临界点。我们描述了这样的集合必须满足的基本性质，然后建立了对集合成员的同调障碍。这些障碍是由由循环和亚循环覆盖空间产生的结不变量决定的。

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Transactions of the American Mathematical Society, Series B

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1.70

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0.00%

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