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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
Twisting, mutation and knot Floer homology 扭转，突变和结花同源
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-08-05 DOI: 10.4171/QT/119
Peter Lambert-Cole
{"title":"Twisting, mutation and knot Floer homology","authors":"Peter Lambert-Cole","doi":"10.4171/QT/119","DOIUrl":"https://doi.org/10.4171/QT/119","url":null,"abstract":"Let $mathcal{L}$ be a knot with a fixed positive crossing and $mathcal{L}_n$ the link obtained by replacing this crossing with $n$ positive twists. We prove that the knot Floer homology $widehat{text{HFK}}(mathcal{L}_n)$ `stabilizes' as $n$ goes to infinity. This categorifies a similar stabilization phenomenon of the Alexander polynomial. As an application, we construct an infinite family of prime, positive mutant knots with isomorphic bigraded knot Floer homology groups. Moreover, given any pair of positive mutants, we describe how to derive a corresponding infinite family positive mutants with isomorphic bigraded $widehat{text{HFK}}$ groups, Seifert genera, and concordance invariant $tau$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"4 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84736980","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
Classification of Thurston relation subfactor planar algebras Thurston关系子因子平面代数的分类
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-06-02 DOI: 10.4171/qt/126
Corey Jones, Zhengwei Liu, Yunxiang Ren
{"title":"Classification of Thurston relation subfactor planar algebras","authors":"Corey Jones, Zhengwei Liu, Yunxiang Ren","doi":"10.4171/qt/126","DOIUrl":"https://doi.org/10.4171/qt/126","url":null,"abstract":"Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either $E^6$ or the ones from representations of quantum $SU(N)$. We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"90 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86797957","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
Categorification of quantum symmetric pairs I 量子对称对的分类1
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-05-12 DOI: 10.4171/QT/117
Huanchen Bao, P. Shan, Weiqiang Wang, Ben Webster
{"title":"Categorification of quantum symmetric pairs I","authors":"Huanchen Bao, P. Shan, Weiqiang Wang, Ben Webster","doi":"10.4171/QT/117","DOIUrl":"https://doi.org/10.4171/QT/117","url":null,"abstract":"We categorify a coideal subalgebra of the quantum group of $mathfrak{sl}_{2r+1}$ by introducing a $2$-category a la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $mathcal{O}$ of type B/C.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2016-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83211314","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
Skein and cluster algebras of marked surfaces 标记曲面的束代数和簇代数
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-01-01 DOI: 10.4171/QT/79
G. Muller
{"title":"Skein and cluster algebras of marked surfaces","authors":"G. Muller","doi":"10.4171/QT/79","DOIUrl":"https://doi.org/10.4171/QT/79","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"76 1","pages":"435-503"},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89011928","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: 63
On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot $5_2$结Kashaev不变量的渐近展开式
IF 1.1 2区 数学
Quantum Topology Pub Date : 2016-01-01 DOI: 10.4171/QT/83
T. Ohtsuki
{"title":"On the asymptotic expansion of the Kashaev invariant of the $5_2$ knot","authors":"T. Ohtsuki","doi":"10.4171/QT/83","DOIUrl":"https://doi.org/10.4171/QT/83","url":null,"abstract":"We give a presentation of the asymptotic expansion of the Kashaev invariant of the 52 knot. As the volume conjecture states, the leading term of the expansion presents the hyperbolic volume and the Chern-Simons invariant of the complement of the 52 knot. Further, we obtain a method to compute the full Poincare asymptotics to all orders of the Kashaev invariant of the 52 knot.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"66 1","pages":"669-735"},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86420733","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: 42
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