In this note, we revisit the $Theta$-invariant as defined by R. Bott and the first author. The $Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The $Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $Theta$-invariant that we can define even if the cohomology group is not vanishing.
{"title":"A note on the $Theta$-invariant of 3-manifolds","authors":"A. Cattaneo, Tatsuro Shimizu","doi":"10.4171/QT/146","DOIUrl":"https://doi.org/10.4171/QT/146","url":null,"abstract":"In this note, we revisit the $Theta$-invariant as defined by R. Bott and the first author. The $Theta$-invariant is an invariant of rational homology 3-spheres with acyclic orthogonal local systems, which is a generalization of the 2-loop term of the Chern-Simons perturbation theory. The $Theta$-invariant can be defined when a cohomology group is vanishing. In this note, we give a slightly modified version of the $Theta$-invariant that we can define even if the cohomology group is not vanishing.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82689333","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}
A. Alekseev, Nariya Kawazumi, Y. Kuno, Florian Naef
Let $Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F in {rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $mathfrak{g}(Sigma)$ and its associated graded ${rm gr}, mathfrak{g}(Sigma)$. In this paper, we prove the converse: if $F$ induces an isomorphism $mathfrak{g}(Sigma) cong {rm gr} , mathfrak{g}(Sigma)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.
{"title":"Goldman–Turaev formality implies Kashiwara–Vergne","authors":"A. Alekseev, Nariya Kawazumi, Y. Kuno, Florian Naef","doi":"10.4171/qt/143","DOIUrl":"https://doi.org/10.4171/qt/143","url":null,"abstract":"Let $Sigma$ be a compact connected oriented 2-dimensional manifold with non-empty boundary. In our previous work, we have shown that the solution of generalized (higher genus) Kashiwara-Vergne equations for an automorphism $F in {rm Aut}(L)$ of a free Lie algebra implies an isomorphism between the Goldman-Turaev Lie bialgebra $mathfrak{g}(Sigma)$ and its associated graded ${rm gr}, mathfrak{g}(Sigma)$. In this paper, we prove the converse: if $F$ induces an isomorphism $mathfrak{g}(Sigma) cong {rm gr} , mathfrak{g}(Sigma)$, then it satisfies the Kashiwara-Vergne equations up to conjugation. As an application of our results, we compute the degree one non-commutative Poisson cohomology of the Kirillov-Kostant-Souriau double bracket. The main technical tool used in the paper is a novel characterization of conjugacy classes in the free Lie algebra in terms of cyclic words.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85974709","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}
The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead doubles of a knot satisfies the Slope Conjecture and the Strong Slope Conjecture if the original knot does. Additionally, we provide a proof that there are Whitehead doubles which are not adequate.
{"title":"The Strong Slope Conjecture for twisted generalized Whitehead doubles","authors":"K. Baker, Kimihiko Motegi, T. Takata","doi":"10.4171/qt/242","DOIUrl":"https://doi.org/10.4171/qt/242","url":null,"abstract":"The Slope Conjecture proposed by Garoufalidis asserts that the degree of the colored Jones polynomial determines a boundary slope, and its refinement, the Strong Slope Conjecture proposed by Kalfagianni and Tran asserts that the linear term in the degree determines the topology of an essential surface that satisfies the Slope Conjecture. Under certain hypotheses, we show that twisted, generalized Whitehead doubles of a knot satisfies the Slope Conjecture and the Strong Slope Conjecture if the original knot does. Additionally, we provide a proof that there are Whitehead doubles which are not adequate.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81943006","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 $mathcal{C}(mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $mathcal{C}(mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $text{Rep}(H)subsetmathcal{C}(mathfrak{g},k)_text{pt}$. For $mathfrak{g}=mathfrak{so}_5$ we describe the decomposition of $mathcal{C}(mathfrak{g},k)_R^0$ into prime factors explicitly and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $mathcal{C}(mathfrak{so}_5,k)$ and $mathcal{C}(mathfrak{g}_2,k)$ for $kinmathbb{Z}_{geq1}$.
{"title":"Prime decomposition of modular tensor categories of local modules of type D","authors":"Andrew Schopieray","doi":"10.4171/QT/140","DOIUrl":"https://doi.org/10.4171/QT/140","url":null,"abstract":"Let $mathcal{C}(mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $mathcal{C}(mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $text{Rep}(H)subsetmathcal{C}(mathfrak{g},k)_text{pt}$. For $mathfrak{g}=mathfrak{so}_5$ we describe the decomposition of $mathcal{C}(mathfrak{g},k)_R^0$ into prime factors explicitly and as an application we classify relations in the Witt group of nondegenerately braided fusion categories generated by the equivalency classes of $mathcal{C}(mathfrak{so}_5,k)$ and $mathcal{C}(mathfrak{g}_2,k)$ for $kinmathbb{Z}_{geq1}$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82659243","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 a contact 3-manifold $(Y, xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $xi_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(xi_0)in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and N'emethi's lattice cohomology.
{"title":"Heegaard Floer invariants of contact structures on links of surface singularities","authors":"J'ozsef Bodn'ar, O. Plamenevskaya","doi":"10.4171/QT/153","DOIUrl":"https://doi.org/10.4171/QT/153","url":null,"abstract":"Let a contact 3-manifold $(Y, xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $xi_0$. We prove a special property of such contact 3-manifolds of \"algebraic\" origin: the Heegaard Floer invariant $c^+(xi_0)in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's \"height of $U$-tower\" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and N'emethi's lattice cohomology.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85375981","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 prove the unrolled superalgebra $mathcal{U}_{xi}^{H}mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.
{"title":"A Hennings type invariant of 3-manifolds from a topological Hopf superalgebra","authors":"N. Ha","doi":"10.4171/qt/142","DOIUrl":"https://doi.org/10.4171/qt/142","url":null,"abstract":"We prove the unrolled superalgebra $mathcal{U}_{xi}^{H}mathfrak{sl}(2|1)$ has a completion which is a ribbon superalgebra in a topological sense where $xi$ is a root of unity of odd order. Using this ribbon superalgebra we construct its universal invariant of links. We use it to construct an invariant of $3$-manifolds of Hennings type.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90067225","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}
{"title":"Holonomy perturbations and regularity for traceless SU(2) character varieties of tangles","authors":"Christopher Herald, P. Kirk","doi":"10.4171/QT/110","DOIUrl":"https://doi.org/10.4171/QT/110","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77016786","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}
For any natural number $n geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $mathbb{O}_n$.
{"title":"A categorification of cyclotomic rings","authors":"Robert Laugwitz, You Qi","doi":"10.4171/QT/172","DOIUrl":"https://doi.org/10.4171/QT/172","url":null,"abstract":"For any natural number $n geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $mathbb{O}_n$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82198341","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 establish relationships between two classes of invariants of Legendrian knots in $mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $beta subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,beta)}(z)$ of the satellite of $K$ with $beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power, and we use this formula to prove that arbitrary $m$-graded satellite ruling polynomials, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of $K$. Conversely, for $mneq 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to $mathbb{F}_q^n$, $mbox{Rep}_m(K,mathbb{F}_q^n)$, is exactly equal to $R^m_{n,K}(q)$. In the case of $2$-graded representations, we show that $R^2_{n,K}=mbox{Rep}_2(K, mathbb{F}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.
{"title":"Satellite ruling polynomials, DGA representations, and the colored HOMFLY-PT polynomial","authors":"C. Leverson, Dan Rutherford","doi":"10.4171/qt/133","DOIUrl":"https://doi.org/10.4171/qt/133","url":null,"abstract":"We establish relationships between two classes of invariants of Legendrian knots in $mathbb{R}^3$: Representation numbers of the Chekanov-Eliashberg DGA and satellite ruling polynomials. For positive permutation braids, $beta subset J^1S^1$, we give a precise formula in terms of representation numbers for the $m$-graded ruling polynomial $R^m_{S(K,beta)}(z)$ of the satellite of $K$ with $beta$ specialized at $z=q^{1/2}-q^{-1/2}$ with $q$ a prime power, and we use this formula to prove that arbitrary $m$-graded satellite ruling polynomials, $R^m_{S(K,L)}$, are determined by the Chekanov-Eliashberg DGA of $K$. Conversely, for $mneq 1$, we introduce an $n$-colored $m$-graded ruling polynomial, $R^m_{n,K}(q)$, in strict analogy with the $n$-colored HOMFLY-PT polynomial, and show that the total $n$-dimensional $m$-graded representation number of $K$ to $mathbb{F}_q^n$, $mbox{Rep}_m(K,mathbb{F}_q^n)$, is exactly equal to $R^m_{n,K}(q)$. In the case of $2$-graded representations, we show that $R^2_{n,K}=mbox{Rep}_2(K, mathbb{F}_q^n)$ arises as a specialization of the $n$-colored HOMFLY-PT polynomial.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2018-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87939914","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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