由耳环缠结在枕套上引起的对应关系

Pub Date : 2022-11-11 DOI:10.1112/topo.12272

Guillem Cazassus, Christopher Herald, Paul Kirk, Artem Kotelskiy

{"title":"由耳环缠结在枕套上引起的对应关系","authors":"Guillem Cazassus, Christopher Herald, Paul Kirk, Artem Kotelskiy","doi":"10.1112/topo.12272","DOIUrl":null,"url":null,"abstract":"<p>The earring tangle consists of four strands <math>\n <semantics>\n <mrow>\n <mn>4</mn>\n <mtext>pt</mtext>\n <mo>×</mo>\n <mi>I</mi>\n <mo>⊂</mo>\n <msup>\n <mi>S</mi>\n <mn>2</mn>\n </msup>\n <mo>×</mo>\n <mi>I</mi>\n </mrow>\n <annotation>$4\\text{pt} \\times I \\subset S^2 \\times I$</annotation>\n </semantics></math> and one meridian around one of the strands. Equipping this tangle with a nontrivial <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>O</mi>\n <mo>(</mo>\n <mn>3</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$SO(3)$</annotation>\n </semantics></math> bundle, we show that its traceless <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>U</mi>\n <mo>(</mo>\n <mn>2</mn>\n <mo>)</mo>\n </mrow>\n <annotation>$SU(2)$</annotation>\n </semantics></math> flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The correspondence induced on the pillowcase by the earring tangle\",\"authors\":\"Guillem Cazassus, Christopher Herald, Paul Kirk, Artem Kotelskiy\",\"doi\":\"10.1112/topo.12272\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The earring tangle consists of four strands <math>\\n <semantics>\\n <mrow>\\n <mn>4</mn>\\n <mtext>pt</mtext>\\n <mo>×</mo>\\n <mi>I</mi>\\n <mo>⊂</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>2</mn>\\n </msup>\\n <mo>×</mo>\\n <mi>I</mi>\\n </mrow>\\n <annotation>$4\\\\text{pt} \\\\times I \\\\subset S^2 \\\\times I$</annotation>\\n </semantics></math> and one meridian around one of the strands. Equipping this tangle with a nontrivial <math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n <mi>O</mi>\\n <mo>(</mo>\\n <mn>3</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$SO(3)$</annotation>\\n </semantics></math> bundle, we show that its traceless <math>\\n <semantics>\\n <mrow>\\n <mi>S</mi>\\n <mi>U</mi>\\n <mo>(</mo>\\n <mn>2</mn>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$SU(2)$</annotation>\\n </semantics></math> flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2022-11-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/topo.12272\",\"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.12272","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}

引用次数: 2

摘要

耳环缠结由四根线组成4 pt × I∧s2 × I$ 4\text{pt} \乘以I \子集S^2 \乘以I$和一条围绕其中一根线的子午线。用一个非平凡的SO(3)$ SO(3)$束来装备这个缠结，我们证明了它的无迹SU(2)$ SU(2)$平坦模空间在拓扑上是光滑的三格曲面。我们还证明了从该表面到耳环缠结边界的无迹平面模空间的限制映射是两个枕套的特定拉格朗日浸入积。后一种计算表明，数字8冒泡——由Bottman和Wehrheim预测的一种微妙的退化现象——出现在无迹字符变体的背景下。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文

The correspondence induced on the pillowcase by the earring tangle

The earring tangle consists of four strands $4 pt \times I \subset S^{2} \times I$ and one meridian around one of the strands. Equipping this tangle with a nontrivial $S O (3)$ bundle, we show that its traceless $S U (2)$ flat moduli space is topologically a smooth genus three surface. We also show that the restriction map from this surface to the traceless flat moduli space of the boundary of the earring tangle is a particular Lagrangian immersion into the product of two pillowcases. The latter computation suggests that figure eight bubbling — a subtle degeneration phenomenon predicted by Bottman and Wehrheim — appears in the context of traceless character varieties.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助