Kauffman bracket skein modules of small 3-manifolds

IF 1.5 1区 数学 Q1 MATHEMATICS
Renaud Detcherry , Efstratia Kalfagianni , Adam S. Sikora
{"title":"Kauffman bracket skein modules of small 3-manifolds","authors":"Renaud Detcherry ,&nbsp;Efstratia Kalfagianni ,&nbsp;Adam S. Sikora","doi":"10.1016/j.aim.2025.110169","DOIUrl":null,"url":null,"abstract":"<div><div>The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed 3-manifolds are finitely generated over <span><math><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>. In this paper, we develop a novel method for computing these skein modules.</div><div>We show that if the skein module <span><math><mi>S</mi><mo>(</mo><mi>M</mi><mo>,</mo><mi>Q</mi><mo>[</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo><mo>)</mo></math></span> of <em>M</em> is tame (e.g. finitely generated over <span><math><mi>Q</mi><mo>[</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>±</mo><mn>1</mn></mrow></msup><mo>]</mo></math></span>), and the <span><math><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>-character scheme is reduced, then the dimension <span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></msub><mo>⁡</mo><mspace></mspace><mi>S</mi><mo>(</mo><mi>M</mi><mo>,</mo><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>)</mo></math></span> is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating <span><math><msub><mrow><mi>dim</mi></mrow><mrow><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow></msub><mo>⁡</mo><mspace></mspace><mi>S</mi><mo>(</mo><mi>M</mi><mo>,</mo><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>)</mo></math></span> to the Abouzaid-Manolescu <span><math><mi>S</mi><mi>L</mi><mo>(</mo><mn>2</mn><mo>,</mo><mi>C</mi><mo>)</mo></math></span>-Floer theoretic invariants, for infinite families of 3-manifolds.</div><div>We prove a criterion for reducedness of character varieties of closed 3-manifolds and use it to compute the skein modules of Dehn fillings of <span><math><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds.</div><div>We also prove that the skein modules of rational homology spheres have dimension at least 1 over <span><math><mi>Q</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"467 ","pages":"Article 110169"},"PeriodicalIF":1.5000,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870825000672","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed 3-manifolds are finitely generated over Q(A). In this paper, we develop a novel method for computing these skein modules.
We show that if the skein module S(M,Q[A±1]) of M is tame (e.g. finitely generated over Q[A±1]), and the SL(2,C)-character scheme is reduced, then the dimension dimQ(A)S(M,Q(A)) is the number of closed points in this character scheme. This, in particular, verifies a conjecture in the literature relating dimQ(A)S(M,Q(A)) to the Abouzaid-Manolescu SL(2,C)-Floer theoretic invariants, for infinite families of 3-manifolds.
We prove a criterion for reducedness of character varieties of closed 3-manifolds and use it to compute the skein modules of Dehn fillings of (2,2n+1)-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds.
We also prove that the skein modules of rational homology spheres have dimension at least 1 over Q(A).
求助全文
约1分钟内获得全文 求助全文
来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信