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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
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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
Fully extended $r$-spin TQFTs 完全扩展的$r$-spin tqft
2区 数学
Quantum Topology Pub Date : 2023-10-15 DOI: 10.4171/qt/193
Nils Carqueville, Lóránt Szegedy
{"title":"Fully extended $r$-spin TQFTs","authors":"Nils Carqueville, Lóránt Szegedy","doi":"10.4171/qt/193","DOIUrl":"https://doi.org/10.4171/qt/193","url":null,"abstract":"We prove the $r$-spin cobordism hypothesis in the setting of (weak) 2-categories for every positive integer $r$: the 2-groupoid of 2-dimensional fully extended $r$-spin TQFTs with given target is equivalent to the homotopy fixed points of an induced $mathrm{Spin}_2^r$-action. In particular, such TQFTs are classified by fully dualisable objects together with a trivialisation of the $r$th power of their Serre automorphisms. For $r=1$, we recover the oriented case (on which our proof builds), while ordinary spin structures correspond to $r=2$. To construct examples, we explicitly describe $mathrm{Spin}_2^r$-homotopy fixed points in the equivariant completion of any symmetric monoidal 2-category. We also show that every object in a 2-category of Landau–Ginzburg models gives rise to fully extended spin TQFTs and that half of these do not factor through the oriented bordism 2-category.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"238 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136185733","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
The adjoint Reidemeister torsion for the connected sum of knots 结的连通和的伴随Reidemeister扭转
2区 数学
Quantum Topology Pub Date : 2023-09-15 DOI: 10.4171/qt/180
Joan Porti, Seokbeom Yoon
{"title":"The adjoint Reidemeister torsion for the connected sum of knots","authors":"Joan Porti, Seokbeom Yoon","doi":"10.4171/qt/180","DOIUrl":"https://doi.org/10.4171/qt/180","url":null,"abstract":"Let $K$ be the connected sum of knots $K_1,ldots,K_n$. It is known that the $mathrm{SL}_2(mathbb{C})$-character variety of the knot exterior of $K$ has a component of dimension $geq 2$ as the connected sum admits a so-called bending. We show that there is a natural way to define the adjoint Reidemeister torsion for such a high-dimensional component and prove that it is locally constant on a subset of the character variety where the trace of a meridian is constant. We also prove that the adjoint Reidemeister torsion of $K$ satisfies the vanishing identity if each $K_i$ does so.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"176 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135353864","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
Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots 捻结的几何三角剖分和teichm<e:1> ller TQFT体积猜想
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-09-01 DOI: 10.4171/qt/178
Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa
{"title":"Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots","authors":"Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa","doi":"10.4171/qt/178","DOIUrl":"https://doi.org/10.4171/qt/178","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"110 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76007100","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
Triangular decomposition of $mathrm{SL}_3$ skein algebras $ mathm {SL}_3$ skein代数的三角分解
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-06-29 DOI: 10.4171/qt/177
Vijay Higgins
{"title":"Triangular decomposition of $mathrm{SL}_3$ skein algebras","authors":"Vijay Higgins","doi":"10.4171/qt/177","DOIUrl":"https://doi.org/10.4171/qt/177","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"214 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75577123","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
$A_infty$-category of Lagrangian cobordisms in the symplectization of $Ptimes {mathbb R}$ $A_infty$的化简中的拉格朗日协数范畴 $Ptimes {mathbb R}$
IF 1.1 2区 数学
Quantum Topology Pub Date : 2023-06-29 DOI: 10.4171/qt/179
N. Legout
{"title":"$A_infty$-category of Lagrangian cobordisms in the symplectization of $Ptimes {mathbb R}$","authors":"N. Legout","doi":"10.4171/qt/179","DOIUrl":"https://doi.org/10.4171/qt/179","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87396258","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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