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Bordered knot algebras with matchings 带匹配的有边结代数
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-07-03 DOI: 10.4171/qt/127
P. Ozsváth, Z. Szabó
{"title":"Bordered knot algebras with matchings","authors":"P. Ozsváth, Z. Szabó","doi":"10.4171/qt/127","DOIUrl":"https://doi.org/10.4171/qt/127","url":null,"abstract":"This paper generalizes the bordered-algebraic knot invariant introduced in an earlier paper, giving an invariant now with more algebraic structure. It also introduces signs to define these invariants with integral coefficients. We describe effective computations of the resulting invariant.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72944201","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: 29
Ore's theorem on subfactor planar algebras 关于子因子平面代数的奥尔定理
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-04-03 DOI: 10.4171/QT/141
S. Palcoux
{"title":"Ore's theorem on subfactor planar algebras","authors":"S. Palcoux","doi":"10.4171/QT/141","DOIUrl":"https://doi.org/10.4171/QT/141","url":null,"abstract":"This article proves that an irreducible subfactor planar algebra with a distributive biprojection lattice admits a minimal 2-box projection generating the identity biprojection. It is a generalization (conjectured in 2013) of a theorem of Oystein Ore on distributive intervals of finite groups (1938), and a corollary of a natural subfactor extension of a conjecture of Kenneth S. Brown in algebraic combinatorics (2000). We deduce a link between combinatorics and representations in finite group theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"55 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86730633","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: 4
Fukaya categories of plumbings and multiplicative preprojective algebras 管道与乘法预射影代数的Fukaya范畴
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-03-13 DOI: 10.4171/QT/131
Tolga Etgu, Yankı Lekili
{"title":"Fukaya categories of plumbings and multiplicative preprojective algebras","authors":"Tolga Etgu, Yankı Lekili","doi":"10.4171/QT/131","DOIUrl":"https://doi.org/10.4171/QT/131","url":null,"abstract":"Given an arbitrary graph $Gamma$ and non-negative integers $g_v$ for each vertex $v$ of $Gamma$, let $X_Gamma$ be the Weinstein $4$-manifold obtained by plumbing copies of $T^*Sigma_v$ according to this graph, where $Sigma_v$ is a surface of genus $g_v$. We compute the wrapped Fukaya category of $X_Gamma$ (with bulk parameters) using Legendrian surgery extending our previous work arXiv:1502.07922 where it was assumed that $g_v=0$ for all $v$ and $Gamma$ was a tree. The resulting algebra is recognized as the (derived) multiplicative preprojective algebra (and its higher genus version) defined by Crawley-Boevey and Shaw arXiv:math/0404186. Along the way, we find a smaller model for the internal DG-algebra of Ekholm-Ng arXiv:1307.8436 associated to $1$-handles in the Legendrian surgery presentation of Weinstein $4$-manifolds which might be of independent interest.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"69 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81502094","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: 18
How to categorify the ring of integers localized at two 如何对定位于2的整数环进行分类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-02-24 DOI: 10.4171/qt/130
M. Khovanov, Yin Tian
{"title":"How to categorify the ring of integers localized at two","authors":"M. Khovanov, Yin Tian","doi":"10.4171/qt/130","DOIUrl":"https://doi.org/10.4171/qt/130","url":null,"abstract":"We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"299 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77331757","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: 8
The Kontsevich integral for bottom tangles in handlebodies 柄体底缠结的Kontsevich积分
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-02-02 DOI: 10.4171/qt/155
K. Habiro, G. Massuyeau
{"title":"The Kontsevich integral for bottom tangles in handlebodies","authors":"K. Habiro, G. Massuyeau","doi":"10.4171/qt/155","DOIUrl":"https://doi.org/10.4171/qt/155","url":null,"abstract":"Using an extension of the Kontsevich integral to tangles in handlebodies \u0000similar to a construction given by Andersen, Mattes and Reshetikhin, we \u0000construct a functor $Z:mathcal{B}to widehat{mathbb{A}}$, where \u0000$mathcal{B}$ is the category of bottom tangles in handlebodies and \u0000$widehat{mathbb{A}}$ is the degree-completion of the category $mathbb{A}$ of \u0000Jacobi diagrams in handlebodies. As a symmetric monoidal linear category, \u0000$mathbb{A}$ is the linear PROP governing \"Casimir Hopf algebras\", which are \u0000cocommutative Hopf algebras equipped with a primitive invariant symmetric \u00002-tensor. The functor $Z$ induces a canonical isomorphism $hbox{gr}mathcal{B} \u0000cong mathbb{A}$, where $hbox{gr}mathcal{B}$ is the associated graded of the \u0000Vassiliev-Goussarov filtration on $mathcal{B}$. To each Drinfeld associator \u0000$varphi$ we associate a ribbon quasi-Hopf algebra $H_varphi$ in \u0000$hbox{gr}mathcal{B}$, and we prove that the braided Hopf algebra resulting \u0000from $H_varphi$ by \"transmutation\" is precisely the image by $Z$ of a \u0000canonical Hopf algebra in the braided category $mathcal{B}$. Finally, we \u0000explain how $Z$ refines the LMO functor, which is a TQFT-like functor extending \u0000the Le-Murakami-Ohtsuki invariant","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"44 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89362676","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: 12
Irreducibility of quantum representations of mapping class groups with boundary 具有边界的映射类群的量子表示的不可约性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-01-31 DOI: 10.4171/QT/116
T. Koberda, Ramanujan Santharoubane
{"title":"Irreducibility of quantum representations of mapping class groups with boundary","authors":"T. Koberda, Ramanujan Santharoubane","doi":"10.4171/QT/116","DOIUrl":"https://doi.org/10.4171/QT/116","url":null,"abstract":"We prove that the Witten--Reshetikhin--Turaev $mathrm{SU}(2)$ quantum representations of mapping class groups are always irreducible in the case of surfaces equipped with colored banded points, provided that at least one banded point is colored by one. We thus generalize a well--known result due to J. Roberts.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"5 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77567615","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: 9
Turaev–Viro invariants, colored Jones polynomials, and volume Turaev-Viro不变量，有色琼斯多项式和体积
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-01-26 DOI: 10.4171/QT/120
Renaud Detcherry, Efstratia Kalfagianni, Tian Yang
{"title":"Turaev–Viro invariants, colored Jones polynomials, and volume","authors":"Renaud Detcherry, Efstratia Kalfagianni, Tian Yang","doi":"10.4171/QT/120","DOIUrl":"https://doi.org/10.4171/QT/120","url":null,"abstract":"We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomial of the link. As an application we give the first examples for which the volume conjecture of Chen and the third named author,cite{Chen-Yang} is verified. Namely, we show that the asymptotics of the Turaev-Viro invariants of the Figure-eight knot and the Borromean rings complement determine the corresponding hyperbolic volumes. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem not to be predicted by neither the Kashaev-Murakami-Murakami volume conjecture and various of its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify it for all knots with zero simplicial volume. Finally we observe that our simplicial volume conjecture is stable under connect sum and split unions of links.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87880883","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: 25
Link cobordisms and absolute gradings in link Floer homology 连杆花同调中的连杆配合和绝对分级
IF 1.1 2区 数学
Quantum Topology Pub Date : 2017-01-12 DOI: 10.4171/QT/124
Ian Zemke
{"title":"Link cobordisms and absolute gradings in link Floer homology","authors":"Ian Zemke","doi":"10.4171/QT/124","DOIUrl":"https://doi.org/10.4171/QT/124","url":null,"abstract":"We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"76 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2017-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81207300","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: 44
On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology 论Ozsváth与Szabó结花同源性的边界理论的非范畴化
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-11-23 DOI: 10.4171/QT/123
A. Manion
{"title":"On the decategorification of Ozsváth and Szabó's bordered theory for knot Floer homology","authors":"A. Manion","doi":"10.4171/QT/123","DOIUrl":"https://doi.org/10.4171/QT/123","url":null,"abstract":"We relate decategorifications of Ozsv'ath-Szab'o's new bordered theory for knot Floer homology to representations of $mathcal{U}_q(mathfrak{gl}(1|1))$. Specifically, we consider two subalgebras $mathcal{C}_r(n,mathcal{S})$ and $mathcal{C}_l(n,mathcal{S})$ of Ozsv'ath- Szab'o's algebra $mathcal{B}(n,mathcal{S})$, and identify their Grothendieck groups with tensor products of representations $V$ and $V^*$ of $mathcal{U}_q(mathfrak{gl}(1|1))$, where $V$ is the vector representation. We identify the decategorifications of Ozsv'ath-Szab'o's DA bimodules for elementary tangles with corresponding maps between representations. Finally, when the algebras are given multi-Alexander gradings, we demonstrate a relationship between the decategorification of Ozsv'ath-Szab'o's theory and Viro's quantum relative $mathcal{A}^1$ of the Reshetikhin-Turaev functor based on $mathcal{U}_q(mathfrak{gl}(1|1))$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"68 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73760063","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: 18
Skein relations for tangle Floer homology 缠结花同调的绞结关系
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-11-13 DOI: 10.4171/QT/134
I. Petkova, C.-M. Michael Wong
{"title":"Skein relations for tangle Floer homology","authors":"I. Petkova, C.-M. Michael Wong","doi":"10.4171/QT/134","DOIUrl":"https://doi.org/10.4171/QT/134","url":null,"abstract":"In a previous paper, V'ertesi and the first author used grid-like Heegaard diagrams to define tangle Floer homology, which associates to a tangle $T$ a differential graded bimodule $widetilde{mathrm{CT}} (T)$. If $L$ is obtained by gluing together $T_1, dotsc, T_m$, then the knot Floer homology $widehat{mathrm{HFK}}(L)$ of $L$ can be recovered from $widetilde{mathrm{CT}} (T_1), dotsc, widetilde{mathrm{CT}} (T_m)$. In the present paper, we prove combinatorially that tangle Floer homology satisfies unoriented and oriented skein relations, generalizing the skein exact triangles for knot Floer homology.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"322 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74978965","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
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