Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in $I$-bundles over compact surfaces. We prove that for a link $Lsubset (-1,1)times T^2$, the Asaeda-Przytycki-Sikora homology of $L$ has rank $2$ with $mathbb{Z}/2$-coefficients if and only if $L$ is isotopic to an embedded knot in ${0}times T^2$.
{"title":"Instantons and Khovanov skein homology on $Itimes T^{2}$","authors":"Yi Xie, Boyu Zhang","doi":"10.4171/qt/184","DOIUrl":"https://doi.org/10.4171/qt/184","url":null,"abstract":"Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in $I$-bundles over compact surfaces. We prove that for a link $Lsubset (-1,1)times T^2$, the Asaeda-Przytycki-Sikora homology of $L$ has rank $2$ with $mathbb{Z}/2$-coefficients if and only if $L$ is isotopic to an embedded knot in ${0}times T^2$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141202881","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 construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of sl2. From the above we derive a categorification of the blob algebra.
{"title":"Tensor product categorifications, Verma modules and the blob 2-category","authors":"Abel Lacabanne, Gr'egoire Naisse, P. Vaz","doi":"10.4171/QT/156","DOIUrl":"https://doi.org/10.4171/QT/156","url":null,"abstract":"We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of sl2. From the above we derive a categorification of the blob algebra.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87478265","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 reduced $mathrm{SL}_2(mathbb{C})$-twisted Burau representation can be obtained from the quantum group $mathcal{U}_q(mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $mathcal{U}_q(mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $operatorname{SL}_2(mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.
{"title":"Holonomy invariants of links and nonabelian Reidemeister torsion","authors":"Calvin McPhail-Snyder","doi":"10.4171/QT/160","DOIUrl":"https://doi.org/10.4171/QT/160","url":null,"abstract":"We show that the reduced $mathrm{SL}_2(mathbb{C})$-twisted Burau representation can be obtained from the quantum group $mathcal{U}_q(mathfrak{sl}_2)$ for $q = i$ a fourth root of unity and that representations of $mathcal{U}_q(mathfrak{sl}_2)$ satisfy a type of Schur-Weyl duality with the Burau representation. As a consequence, the $operatorname{SL}_2(mathbb{C})$-twisted Reidemeister torsion of links can be obtained as a quantum invariant. Our construction is closely related to the quantum holonomy invariant of Blanchet, Geer, Patureau-Mirand, and Reshetikhin, and we interpret their invariant as a twisted Conway potential.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88601129","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 this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r in Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum $mathfrak{sl}_2$ completion studied in arXiv:math/0605313.
本文证明了$S^3$中一个结的有色琼斯多项式族决定了该结的ADO多项式族。更精确地说，我们构造了一个统一ADO多项式和有色琼斯多项式的双变量结不变量。一方面，第一个变量$q$可以在$r in Bbb N^*$的$2r$单位根处求值，得到了Alexander多项式上的ADO多项式。另一方面，第二个变量$A$在$A=q^n$处求值，给出了有色琼斯多项式。由此，我们展示了一个映射，对于任何结，彩色琼斯多项式族到ADO多项式族。作为这一事实的直接应用，我们将证明每个ADO多项式都是q完整的，并且被与有色琼斯函数相同的多项式所湮灭。统一不变量的构造将使用环和代数的补全。我们还将展示如何从arXiv:math/0605313中研究的Habiro量子$mathfrak{sl}_2$补全中恢复我们的不变量。
{"title":"A unification of the ADO and colored Jones polynomials of a knot","authors":"Sonny Willetts","doi":"10.4171/qt/161","DOIUrl":"https://doi.org/10.4171/qt/161","url":null,"abstract":"In this paper we prove that the family of colored Jones polynomials of a knot in $S^3$ determines the family of ADO polynomials of this knot. More precisely, we construct a two variables knot invariant unifying both the ADO and the colored Jones polynomials. On one hand, the first variable $q$ can be evaluated at $2r$ roots of unity with $r in Bbb N^*$ and we obtain the ADO polynomial over the Alexander polynomial. On the other hand, the second variable $A$ evaluated at $A=q^n$ gives the colored Jones polynomials. From this, we exhibit a map sending, for any knot, the family of colored Jones polynomials to the family of ADO polynomials. As a direct application of this fact, we will prove that every ADO polynomial is q-holonomic and is annihilated by the same polynomials as of the colored Jones function. The construction of the unified invariant will use completions of rings and algebra. We will also show how to recover our invariant from Habiro's quantum $mathfrak{sl}_2$ completion studied in arXiv:math/0605313.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79746984","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 this paper we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain stacky nodal curves, some of whose connected components have non-trivial generic stabiliser.
{"title":"Homological mirror symmetry for invertible polynomials in two variables","authors":"Matthew Habermann","doi":"10.4171/QT/163","DOIUrl":"https://doi.org/10.4171/QT/163","url":null,"abstract":"In this paper we give a proof of homological mirror symmetry for two variable invertible polynomials, where the symmetry group on the $B$-side is taken to be maximal. The proof involves an explicit gluing construction of the Milnor fibres, and as an application, we prove derived equivalences between certain stacky nodal curves, some of whose connected components have non-trivial generic stabiliser.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78328374","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 identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte-Gaiotto-Gukov. This implies the topological invariance of the $q$-series of Frohman and Kania-Bartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte-Gaiotto-Gukov as a limit of quantum 6j-symbols.
{"title":"The FKB invariant is the 3d index","authors":"S. Garoufalidis, R. Veen","doi":"10.4171/qt/171","DOIUrl":"https://doi.org/10.4171/qt/171","url":null,"abstract":"We identify the q-series associated to an 1-efficient ideal triangulation of a cusped hyperbolic 3-manifold by Frohman and Kania-Bartoszynska with the 3D-index of Dimofte-Gaiotto-Gukov. This implies the topological invariance of the $q$-series of Frohman and Kania-Bartoszynska for cusped hyperbolic 3-manifolds. Conversely, we identify the tetrahedron index of Dimofte-Gaiotto-Gukov as a limit of quantum 6j-symbols.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72899437","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 [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of "stringnets with boundary conditions" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].
{"title":"Factorization homology and 4D TQFT","authors":"A. Kirillov, Ying Hong Tham","doi":"10.4171/QT/159","DOIUrl":"https://doi.org/10.4171/QT/159","url":null,"abstract":"In [BK], it is shown that the Turaev-Viro invariants defined for a spherical fusion category $mathcal{A}$ extends to invariants of 3-manifolds with corners. In [Kir], an equivalent formulation for the 2-1 part of the theory (2-manifolds with boundary) is described using the space of \"stringnets with boundary conditions\" as the vector spaces associated to 2-manifolds with boundary. Here we construct a similar theory for the 3-2 part of the 4-3-2 theory in [CY1993].","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84451492","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 any pivotal Hopf monoid $H$ in a symmetric monoidal category $mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g geq 1$ with $n geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.
{"title":"Mapping class group actions from Hopf monoids and ribbon graphs","authors":"C. Meusburger, T. Voss","doi":"10.4171/QT/158","DOIUrl":"https://doi.org/10.4171/QT/158","url":null,"abstract":"We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g geq 1$ with $n geq 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter-Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter-Drinfeld module structure.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76610518","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 Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $mathbb{A}$ in a modular fusion category $mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $mathbb{A}$ in $mathcal{C}$, we introduce a ribbon category $mathcal{C}_{mathbb{A}}$ and show that it is again a modular fusion category. The definition of $mathcal{C}_{mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $mathbb{A}$ is given by a simple commutative $Delta$-separable Frobenius algebra $A$ in $mathcal{C}$; (ii) when $mathbb{A}$ is an orbifold datum in $mathcal{C} = operatorname{Vect}$, built from a spherical fusion category $mathcal{S}$. We show that in case (i), $mathcal{C}_{mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $mathcal{S}$. The category $mathcal{C}_{mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.
在Carqueville et al.， arXiv:1809.01483中，作为Reshetikhin-Turaev tqft的广义轨道构造的一部分，引入了模融合范畴$mathcal{A}$中的轨道基准$mathbb{A}$的概念。本文给出了$mathcal{C}$中的一个简单的轨道基准$mathbb{a}$，引入了一个带状范畴$mathcal{C}_{mathbb{a}}$，并证明了它也是一个模融合范畴。$mathcal{C}_{mathbb{A}}$的定义是由广义轨道折中的Wilson线的性质所激发的。我们详细地分析了两个例子:(i)当$mathbb{A}$由一个简单交换$Delta$-可分Frobenius代数$A$在$mathcal{C}$中给出;(ii)当$mathbb{A}$是$mathcal{C} = operatorname{Vect}$中的一个轨道基准时，从一个球面融合类别$mathcal{S}$中构建。我们证明了在情形(i)下，$mathcal{C}_{mathbb{A}}$与$A$的局部模的范畴是带状等价的，在情形(ii)下，与$mathcal{S}$的德林菲尔德中心是带状等价的。范畴$mathcal{C}_{mathbb{A}}$因此将这两个结构统一为一个代数设置。
{"title":"Constructing modular categories from orbifold data","authors":"Vincentas Mulevičius, I. Runkel","doi":"10.4171/qt/170","DOIUrl":"https://doi.org/10.4171/qt/170","url":null,"abstract":"In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $mathbb{A}$ in a modular fusion category $mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $mathbb{A}$ in $mathcal{C}$, we introduce a ribbon category $mathcal{C}_{mathbb{A}}$ and show that it is again a modular fusion category. The definition of $mathcal{C}_{mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $mathbb{A}$ is given by a simple commutative $Delta$-separable Frobenius algebra $A$ in $mathcal{C}$; (ii) when $mathbb{A}$ is an orbifold datum in $mathcal{C} = operatorname{Vect}$, built from a spherical fusion category $mathcal{S}$. We show that in case (i), $mathcal{C}_{mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $mathcal{S}$. The category $mathcal{C}_{mathbb{A}}$ thus unifies these two constructions into a single algebraic setting.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88325039","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 study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. Additionally, we study the Legendrian invariants of these knots in the minus version of the knot Floer homology, obtaining that $Ucdotmathfrak L(L)$ vanishes for any Legendrian negative torus knot $L$ in any overtwisted structure, and that the strongly non-loose transverse realizations $T$ are characterized by having non-zero invariant $mathfrak T(T)$. Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.
{"title":"Non-loose negative torus knots","authors":"Irena Matkovič","doi":"10.4171/qt/169","DOIUrl":"https://doi.org/10.4171/qt/169","url":null,"abstract":"We study Legendrian and transverse realizations of the negative torus knots $T_{(p,-q)}$ in all contact structures on the $3$-sphere. We give a complete classification of the strongly non-loose transverse realizations and the strongly non-loose Legendrian realizations with the Thurston-Bennequin invariant smaller than $-pq$. \u0000Additionally, we study the Legendrian invariants of these knots in the minus version of the knot Floer homology, obtaining that $Ucdotmathfrak L(L)$ vanishes for any Legendrian negative torus knot $L$ in any overtwisted structure, and that the strongly non-loose transverse realizations $T$ are characterized by having non-zero invariant $mathfrak T(T)$. \u0000Along the way, we relate our Legendrian realizations to the tight contact structures on the Legendrian surgeries along them. Specifically, we realize all tight structures on the lens spaces $L(pq+1,p^2)$ as a single Legendrian surgery on a Legendrian $T_{(p,-q)}$, and we relate transverse realizations in overtwisted structures to the non-fillable tight structures on the large negative surgeries along the underlying knots.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75931620","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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