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Decategorified Heegaard Floer theory and actions of both $E$ and $F$ 十分类 Heegaard Floer 理论与 $E$ 和 $F$ 的作用
IF 1.1 2区 数学
Quantum Topology Pub Date : 2024-01-08 DOI: 10.4171/qt/204
Andrew Manion
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引用次数: 0
Curves in the disc, the type $B$ braid group, and a type $B$ zigzag algebra 圆盘中的曲线、B$型辫状群和B$型之字形代数
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-11-22 DOI: 10.4171/qt/198
Edmund Heng, Kie Seng Nge
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引用次数: 0
Invariants of $mathbb{Z}/p$-homology 3-spheres from the abelianization of the level-$p$ mapping class group 从水平-$p$映射类群的无差别化看$mathbb{Z}/p$-homology 3-球体的不变式
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-11-17 DOI: 10.4171/qt/196
Wolfgang Pitsch, Ricard Riba
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引用次数: 0
Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles 用3柄Kirby图求$G$交叉编织球融合范畴的TQFT不变量
2区 数学
Quantum Topology Pub Date : 2023-11-14 DOI: 10.4171/qt/183
Manuel Bärenz
{"title":"Evaluating TQFT invariants from $G$-crossed braided spherical fusion categories via Kirby diagrams with 3-handles","authors":"Manuel Bärenz","doi":"10.4171/qt/183","DOIUrl":"https://doi.org/10.4171/qt/183","url":null,"abstract":"A family of TQFTs parametrised by G-crossed braided spherical fusion categories has been defined recently as a state sum model and as a Hamiltonian lattice model. Concrete calculations of the resulting manifold invariants are scarce because of the combinatorial complexity of triangulations, if nothing else. Handle decompositions, and in particular Kirby diagrams are known to offer an economic and intuitive description of 4-manifolds. We show that if 3-handles are added to the picture, the state sum model can be conveniently redefined by translating Kirby diagrams into the graphical calculus of a G-crossed braided spherical fusion category. ","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"5 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229844","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Actions of $sltwo$ on algebras appearing in categorification $sltwo$对分类中出现的代数的作用
2区 数学
Quantum Topology Pub Date : 2023-11-13 DOI: 10.4171/qt/181
Ben Elias, You Qi
{"title":"Actions of $sltwo$ on algebras appearing in categorification","authors":"Ben Elias, You Qi","doi":"10.4171/qt/181","DOIUrl":"https://doi.org/10.4171/qt/181","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"26 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136282070","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On $mathfrak{sl}(N)$ link homology with mod $N$ coefficients 关于$mathfrak{sl}(N)$与mod $N$系数的链接同调
2区 数学
Quantum Topology Pub Date : 2023-11-03 DOI: 10.4171/qt/194
Joshua Wang
{"title":"On $mathfrak{sl}(N)$ link homology with mod $N$ coefficients","authors":"Joshua Wang","doi":"10.4171/qt/194","DOIUrl":"https://doi.org/10.4171/qt/194","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"114 4‐5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Quantized representations of knot groups 结群的量化表示
2区 数学
Quantum Topology Pub Date : 2023-11-03 DOI: 10.4171/qt/191
Jun Murakami, Roland van der Veen
{"title":"Quantized representations of knot groups","authors":"Jun Murakami, Roland van der Veen","doi":"10.4171/qt/191","DOIUrl":"https://doi.org/10.4171/qt/191","url":null,"abstract":"We propose a new non-commutative generalization of the representation variety and the character variety of a knot group. Our strategy is to reformulate the construction of the algebra of functions on the space of representations in terms of Hopf algebra objects in a braided category (braided Hopf algebra). The construction works under the assumption that the algebra is braided commutative. The resulting knot invariant is a module with a coadjoint action. Taking the coinvariants yields a new quantum character variety that may be thought of as an alternative to the skein module. We give concrete examples for a few of the simplest knots and links.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"113 2‐3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Witten–Reshetikhin–Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces 构型空间中拉格朗日交点上3流形的Witten-Reshetikhin-Turaev不变量
2区 数学
Quantum Topology Pub Date : 2023-11-01 DOI: 10.4171/qt/190
Cristina Ana-Maria Anghel
{"title":"Witten–Reshetikhin–Turaev invariants for 3-manifolds from Lagrangian intersections in configuration spaces","authors":"Cristina Ana-Maria Anghel","doi":"10.4171/qt/190","DOIUrl":"https://doi.org/10.4171/qt/190","url":null,"abstract":"In this paper we construct a topological model for the Witten-Reshetikhin-Turaev invariants for $3$-manifolds coming from the quantum group $U_q(sl(2))$, as graded intersection pairings of homology classes in configuration spaces. More precisely, for a fixed level $cN in N$ we show that the level $cN$ WRT invariant for a $3-$manifold is a state sum of Lagrangian intersections in a covering of a {bf fixed} configuration space in the punctured disk. This model brings a new perspective on the structure of the level $cN$ Witten-Reshetikhin-Turaev invariant, showing that it is completely encoded by the intersection points between certain Lagrangian submanifolds in a fixed configuration space, with additional gradings which come from a particular choice of a local system. This formula provides a new framework for investigating the open question about categorifications of the WRT invariants.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"41 10","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136103393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Augmented Legendrian cobordism in $J^1S^1$ $J^1S^1$中的增广legendard协同
2区 数学
Quantum Topology Pub Date : 2023-10-25 DOI: 10.4171/qt/195
Yu Pan, Dan Rutherford
{"title":"Augmented Legendrian cobordism in $J^1S^1$","authors":"Yu Pan, Dan Rutherford","doi":"10.4171/qt/195","DOIUrl":"https://doi.org/10.4171/qt/195","url":null,"abstract":"We consider Legendrian links and tangles in $J^1S^1$ and $J^1[0,1]$ equipped with Morse complex families over a field $mathbb{F}$ and classify them up to Legendrian cobordism. When the coefficient field is $mathbb{F}_2$, this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod $2$ spin number. We apply the classification to construct augmented Legendrian surfaces in $J^1M$ with $mathrm{dim} M = 2$ realizing any prescribed monodromy representation, $Phi:pi_1(M,x_0) to mathrm{GL}(mathbf{n}, mathbb{F})$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134973344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Mapping class group representations and Morita classes of algebras 代数的映射类群表示和森田类
2区 数学
Quantum Topology Pub Date : 2023-10-19 DOI: 10.4171/qt/192
Iordanis Romaidis, Ingo Runkel
{"title":"Mapping class group representations and Morita classes of algebras","authors":"Iordanis Romaidis, Ingo Runkel","doi":"10.4171/qt/192","DOIUrl":"https://doi.org/10.4171/qt/192","url":null,"abstract":"A modular fusion category $mathcal{C}$ allows one to define projective representations of the mapping class groups of closed surfaces of any genus. We show that if all these representations are irreducible, then $mathcal{C}$ has a unique Morita class of simple non-degenerate algebras, namely, that of the tensor unit. This improves on a result by Andersen and Fjelstad, albeit under stronger assumptions. One motivation to look at this problem comes from questions in three-dimensional quantum gravity.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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