On Floer minimal knots in sutured manifolds

Transactions of the American Mathematical Society, Series B Pub Date : 2021-08-24 DOI:10.1090/btran/105

Zhenkun Li, Yi Xie, Boyu Zhang

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It is known that the rank of the sutured Floer homology of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M minus upper N left-parenthesis upper K right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>M</mml:mi>\n <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">∖</mml:mi>\n <mml:mi>N</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mi>K</mml:mi>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">M\\backslash N(K)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is at least twice the rank of the sutured Floer homology of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper M\">\n <mml:semantics>\n <mml:mi>M</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">M</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. This paper studies the properties of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> when the equality is achieved for instanton homology. As an application, we show that if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L subset-of upper S cubed\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo>⊂</mml:mo>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L\\subset S^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a fixed link and <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is a knot in the complement of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L\">\n <mml:semantics>\n <mml:mi>L</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, then the instanton link Floer homology of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper L union upper K\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>L</mml:mi>\n <mml:mo>∪</mml:mo>\n <mml:mi>K</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">L\\cup K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> achieves the minimum rank if and only if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\">\n <mml:semantics>\n <mml:mi>K</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">K</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is the unknot in <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed minus upper L\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">∖</mml:mi>\n <mml:mi>L</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">S^3\\backslash L</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>.</p>","PeriodicalId":377306,"journal":{"name":"Transactions of the American Mathematical Society, Series B","volume":"79 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the American Mathematical Society, Series B","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/btran/105","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 3

Abstract

Suppose ( M , γ ) (M, \gamma ) is a balanced sutured manifold and K K is a rationally null-homologous knot in M M . It is known that the rank of the sutured Floer homology of M ∖ N ( K ) M\backslash N(K) is at least twice the rank of the sutured Floer homology of M M . This paper studies the properties of K K when the equality is achieved for instanton homology. As an application, we show that if L ⊂ S 3 L\subset S^3 is a fixed link and K K is a knot in the complement of L L , then the instanton link Floer homology of L ∪ K L\cup K achieves the minimum rank if and only if K K is the unknot in S 3 ∖ L S^3\backslash L .

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在花上最小的结在缝合的流形

假设(M， γ) (M, \gamma)是一个平衡的缝合流形，K K是M M中的一个合理的零同源结。已知M∈N(K) M \backslash N(K)的缝合花同调的秩至少是M的缝合花同调的秩的两倍。本文研究了K K在满足实例同调的等式时的性质。作为一个应用，我们证明了如果L∧S 3 L \subset S^3是一个固定的链路，K K是L L补上的一个结，那么L∪K L \cup K的瞬时链路花同调达到最小秩当且仅当K K是S 3∈L S^3 \backslash L中的解结。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Transactions of the American Mathematical Society, Series B

CiteScore

1.70

自引率

0.00%

发文量