Renaud Detcherry, Efstratia Kalfagianni, Tian Yang
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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 $mathcal{U}_q(mathfrak{gl}(1|1))$.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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.
{"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":null,"pages":null},"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}
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 $vec{F}$.
{"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":null,"pages":null},"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}
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.
我们对出现在对应于Temperley-Lieb代数$text{TL}_k(d)$的Kauffman图基的对偶基中的结构系数导出了一个Laurent级数展开式，该展开式收敛于所有复杂回路参数$d$与$|d| > 2cosbig(frac{pi}{k+1}big)$。特别地，这产生了$text{TL}_k(d)$中Jones-Wenzl投影的结构系数的新公式。在每个劳伦展开式中出现的系数被证明有一个自然的组合解释，根据我们在非交叉配对上放置的某个图结构，这些系数证明具有一个显著的性质，即它们要么总是正整数，要么总是负整数。作为应用，我们肯定地回答了Vaughan Jones的一个问题，即在$text{TL}_k(d)$(当$d in mathbb R backslash [-2cosbig(frac{pi}{k+1}big),2cosbig(frac{pi}{k+1}big)]$)的每个对偶基元展开中是否每个Temperley-Lieb图都以非零系数出现。专门研究Jones-Wenzl投影，这个结果给出了Ocneanu结果的一个新的证明，说明在Jones-Wenzl投影中，每个Temperley-Lieb图都以非零系数出现。我们的方法建立了与自由量子群上的Weingarten微积分的联系，并作为副产品得到了自由正交Weingarten函数的改进渐近性。
{"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":null,"pages":null},"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}
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 $tau$.
{"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":null,"pages":null},"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}
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