{"title":"论瞬子和单极浮子理论中的陶不变式","authors":"Sudipta Ghosh, Zhenkun Li, C.-M. Michael Wong","doi":"10.1112/topo.12346","DOIUrl":null,"url":null,"abstract":"<p>We unify two existing approaches to the <i>tau</i> invariants in instanton and monopole Floer theories, by identifying <span></span><math>\n <semantics>\n <msub>\n <mi>τ</mi>\n <mi>G</mi>\n </msub>\n <annotation>$\\tau _{\\mathrm{G}}$</annotation>\n </semantics></math>, defined by the second author via the <i>minus</i> flavors <span></span><math>\n <semantics>\n <msup>\n <munder>\n <mo>KHI</mo>\n <mo>̲</mo>\n </munder>\n <mo>−</mo>\n </msup>\n <annotation>$\\underline{\\operatorname{KHI}}^-$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msup>\n <munder>\n <mo>KHM</mo>\n <mo>̲</mo>\n </munder>\n <mo>−</mo>\n </msup>\n <annotation>$\\underline{\\operatorname{KHM}}^-$</annotation>\n </semantics></math> of the knot homologies, with <span></span><math>\n <semantics>\n <msubsup>\n <mi>τ</mi>\n <mi>G</mi>\n <mo>♯</mo>\n </msubsup>\n <annotation>$\\tau ^\\sharp _{\\mathrm{G}}$</annotation>\n </semantics></math>, defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute <span></span><math>\n <semantics>\n <msup>\n <munder>\n <mo>KHI</mo>\n <mo>̲</mo>\n </munder>\n <mo>−</mo>\n </msup>\n <annotation>$\\underline{\\operatorname{KHI}}^-$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <msup>\n <munder>\n <mo>KHM</mo>\n <mo>̲</mo>\n </munder>\n <mo>−</mo>\n </msup>\n <annotation>$\\underline{\\operatorname{KHM}}^-$</annotation>\n </semantics></math> for twist knots.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12346","citationCount":"0","resultStr":"{\"title\":\"On the tau invariants in instanton and monopole Floer theories\",\"authors\":\"Sudipta Ghosh, Zhenkun Li, C.-M. Michael Wong\",\"doi\":\"10.1112/topo.12346\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We unify two existing approaches to the <i>tau</i> invariants in instanton and monopole Floer theories, by identifying <span></span><math>\\n <semantics>\\n <msub>\\n <mi>τ</mi>\\n <mi>G</mi>\\n </msub>\\n <annotation>$\\\\tau _{\\\\mathrm{G}}$</annotation>\\n </semantics></math>, defined by the second author via the <i>minus</i> flavors <span></span><math>\\n <semantics>\\n <msup>\\n <munder>\\n <mo>KHI</mo>\\n <mo>̲</mo>\\n </munder>\\n <mo>−</mo>\\n </msup>\\n <annotation>$\\\\underline{\\\\operatorname{KHI}}^-$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <msup>\\n <munder>\\n <mo>KHM</mo>\\n <mo>̲</mo>\\n </munder>\\n <mo>−</mo>\\n </msup>\\n <annotation>$\\\\underline{\\\\operatorname{KHM}}^-$</annotation>\\n </semantics></math> of the knot homologies, with <span></span><math>\\n <semantics>\\n <msubsup>\\n <mi>τ</mi>\\n <mi>G</mi>\\n <mo>♯</mo>\\n </msubsup>\\n <annotation>$\\\\tau ^\\\\sharp _{\\\\mathrm{G}}$</annotation>\\n </semantics></math>, defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute <span></span><math>\\n <semantics>\\n <msup>\\n <munder>\\n <mo>KHI</mo>\\n <mo>̲</mo>\\n </munder>\\n <mo>−</mo>\\n </msup>\\n <annotation>$\\\\underline{\\\\operatorname{KHI}}^-$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <msup>\\n <munder>\\n <mo>KHM</mo>\\n <mo>̲</mo>\\n </munder>\\n <mo>−</mo>\\n </msup>\\n <annotation>$\\\\underline{\\\\operatorname{KHM}}^-$</annotation>\\n </semantics></math> for twist knots.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12346\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/topo.12346\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12346","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the tau invariants in instanton and monopole Floer theories
We unify two existing approaches to the tau invariants in instanton and monopole Floer theories, by identifying , defined by the second author via the minus flavors and of the knot homologies, with , defined by Baldwin and Sivek via cobordism maps of the 3-manifold homologies induced by knot surgeries. We exhibit several consequences, including a relationship with Heegaard Floer theory, and use our result to compute and for twist knots.
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