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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
The Khovanov homology of infinite braids 无限辫的Khovanov同调
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-10-14 DOI: 10.4171/QT/114
Gabriel Islambouli, Michael Willis
{"title":"The Khovanov homology of infinite braids","authors":"Gabriel Islambouli, Michael Willis","doi":"10.4171/QT/114","DOIUrl":"https://doi.org/10.4171/QT/114","url":null,"abstract":"We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones-Wenzl projector. This result extends Lev Rozansky's categorification of the Jones-Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz-Sarkar-Khovanov homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"45 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75632556","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: 7
The classification of $3^n$ subfactors and related fusion categories $3^n$子因子的分类及相关的融合范畴
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-09-24 DOI: 10.4171/QT/113
Masaki Izumi
{"title":"The classification of $3^n$ subfactors and related fusion categories","authors":"Masaki Izumi","doi":"10.4171/QT/113","DOIUrl":"https://doi.org/10.4171/QT/113","url":null,"abstract":"We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd $n$.In particular, our method with $n=4$ gives a uniform construction of 4 finite depth subfactors, up to dual,without intermediate subfactors of index $3+sqrt{5}$. It also provides a key step for a new construction of the Asaeda-Haagerup subfactor due to Grossman, Snyder, and the author.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"2012 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86387348","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: 15
Triangular decomposition of skein algebras skein代数的三角分解
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-09-16 DOI: 10.4171/QT/115
Thang T. Q. Lê
{"title":"Triangular decomposition of skein algebras","authors":"Thang T. Q. Lê","doi":"10.4171/QT/115","DOIUrl":"https://doi.org/10.4171/QT/115","url":null,"abstract":"By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"12 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85161359","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: 51
The Homflypt polynomial and the oriented Thompson group Homflypt多项式与取向汤普森群
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-09-08 DOI: 10.4171/QT/112
Valeriano Aiello, R. Conti, V. Jones
{"title":"The Homflypt polynomial and the oriented Thompson group","authors":"Valeriano Aiello, R. Conti, V. Jones","doi":"10.4171/QT/112","DOIUrl":"https://doi.org/10.4171/QT/112","url":null,"abstract":"We show how to construct unitary representations of the oriented Thompson group $vec{F}$ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $vec{F}$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"41 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81278100","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: 19
Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones 坦波利-里布代数中的对偶基，量子群，以及琼斯的一个问题
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-08-12 DOI: 10.4171/QT/118
Michael Brannan, B. Collins
{"title":"Dual bases in Temperley–Lieb algebras, quantum groups, and a question of Jones","authors":"Michael Brannan, B. Collins","doi":"10.4171/QT/118","DOIUrl":"https://doi.org/10.4171/QT/118","url":null,"abstract":"We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $text{TL}_k(d)$, converging for all complex loop parameters $d$ with $|d| > 2cosbig(frac{pi}{k+1}big)$. In particular, this yields a new formula for the structure coefficients of the Jones-Wenzl projection in $text{TL}_k(d)$. The coefficients appearing in each Laurent expansion are shown to have a natural combinatorial interpretation in terms of a certain graph structure we place on non-crossing pairings, and these coefficients turn out to have the remarkable property that they either always positive integers or always negative integers. As an application, we answer affirmatively a question of Vaughan Jones, asking whether every Temperley-Lieb diagram appears with non-zero coefficient in the expansion of each dual basis element in $text{TL}_k(d)$ (when $d in mathbb R backslash [-2cosbig(frac{pi}{k+1}big),2cosbig(frac{pi}{k+1}big)]$). Specializing to Jones-Wenzl projections, this result gives a new proof of a result of Ocneanu, stating that every Temperley-Lieb diagram appears with non-zero coefficient in a Jones-Wenzl projection. Our methods establish a connection with the Weingarten calculus on free quantum groups, and yield as a byproduct improved asymptotics for the free orthogonal Weingarten function.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"14 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83015654","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
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