Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa
{"title":"Geometric triangulations and the Teichmüller TQFT volume conjecture for twist knots","authors":"Fathi Ben Aribi, François Guéritaud, Eiichi Piguet-Nakazawa","doi":"10.4171/qt/178","DOIUrl":"https://doi.org/10.4171/qt/178","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76007100","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":"$A_infty$-category of Lagrangian cobordisms in the symplectization of $Ptimes {mathbb R}$","authors":"N. Legout","doi":"10.4171/qt/179","DOIUrl":"https://doi.org/10.4171/qt/179","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87396258","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 $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)le1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $widehat{HFK}(Z,K,[F],g)$ is supported in a single $mathbb Z/2mathbb Z$--grading, then [mathrm{rank}widehat{HFK}(Z,K,[F],g-1)gemathrm{rank}widehat{HFK}(Z,K,[F],g).]
{"title":"The next-to-top term in knot Floer homology","authors":"Yi Ni","doi":"10.4171/qt/174","DOIUrl":"https://doi.org/10.4171/qt/174","url":null,"abstract":"Let $K$ be a null-homologous knot in a generalized L-space $Z$ with $b_1(Z)le1$. Let $F$ be a Seifert surface of $K$ with genus $g$. We show that if $widehat{HFK}(Z,K,[F],g)$ is supported in a single $mathbb Z/2mathbb Z$--grading, then [mathrm{rank}widehat{HFK}(Z,K,[F],g-1)gemathrm{rank}widehat{HFK}(Z,K,[F],g).]","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79137376","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}
This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $chi_{rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $chi_{rm gr}(SHI(M,gamma))=chi_{rm gr}(SFH(M,gamma))$ for any balanced sutured manifold $(M,gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $chi_{rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $underline{rm KHI}^-(Y,K)$ introduced by the first author.
{"title":"Instanton Floer homology, sutures, and Euler characteristics","authors":"Zhenkun Li, Fan Ye","doi":"10.4171/qt/182","DOIUrl":"https://doi.org/10.4171/qt/182","url":null,"abstract":"This is a companion paper to an earlier work of the authors. In this paper, we provide an axiomatic definition of Floer homology for balanced sutured manifolds and prove that the graded Euler characteristic $chi_{rm gr}$ of this homology is fully determined by the axioms we proposed. As a result, we conclude that $chi_{rm gr}(SHI(M,gamma))=chi_{rm gr}(SFH(M,gamma))$ for any balanced sutured manifold $(M,gamma)$. In particular, for any link $L$ in $S^3$, the Euler characteristic $chi_{rm gr}(KHI(S^3,L))$ recovers the multi-variable Alexander polynomial of $L$, which generalizes the knot case. Combined with the authors' earlier work, we provide more examples of $(1,1)$-knots in lens spaces whose $KHI$ and $widehat{HFK}$ have the same dimension. Moreover, for a rationally null-homologous knot in a closed oriented 3-manifold $Y$, we construct canonical $mathbb{Z}_2$-gradings on $KHI(Y,K)$, the decomposition of $I^sharp(Y)$ discussed in the previous paper, and the minus version of instanton knot homology $underline{rm KHI}^-(Y,K)$ introduced by the first author.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2021-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86984989","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 the asymptotic behavior of the Witten-Reshetikhin-Turaev invariant associated with the square of the $n$-th root of unity with odd $n$ for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern-Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.
{"title":"Quantum invariants of three-manifolds obtained by surgeries along torus knots","authors":"H. Murakami, Anh T. Tran","doi":"10.4171/qt/175","DOIUrl":"https://doi.org/10.4171/qt/175","url":null,"abstract":"We study the asymptotic behavior of the Witten-Reshetikhin-Turaev invariant associated with the square of the $n$-th root of unity with odd $n$ for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern-Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90409169","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}
Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a "light leaves" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $mathfrak{so}_5cong mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_qne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $mathcal{U}_q^{mathbb{Z}}(mathfrak{sp}_4)$.
{"title":"Web calculus and tilting modules in type $C_2$","authors":"Elijah Bodish","doi":"10.4171/qt/166","DOIUrl":"https://doi.org/10.4171/qt/166","url":null,"abstract":"Using Kuperberg's $B_2/C_2$ webs, and following Elias and Libedinsky, we describe a \"light leaves\" algorithm to construct a basis of morphisms between arbitrary tensor products of fundamental representations for $mathfrak{so}_5cong mathfrak{sp}_4$ (and the associated quantum group). Our argument has very little dependence on the base field. As a result, we prove that when $[2]_qne 0$, the Karoubi envelope of the $C_2$ web category is equivalent to the category of tilting modules for the divided powers quantum group $mathcal{U}_q^{mathbb{Z}}(mathfrak{sp}_4)$.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74067365","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":"A closed formula for the evaluation of foams","authors":"Louis-Hadrien Robert, E. Wagner","doi":"10.4171/qt/139","DOIUrl":"https://doi.org/10.4171/qt/139","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77029449","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 recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.
{"title":"Non-semisimple 3-manifold invariants derived from the Kauffman bracket","authors":"M. Renzi, J. Murakami","doi":"10.4171/QT/164","DOIUrl":"https://doi.org/10.4171/QT/164","url":null,"abstract":"We recover the family of non-semisimple quantum invariants of closed oriented 3-manifolds associated with the small quantum group of $mathfrak{sl}_2$ using purely combinatorial methods based on Temperley-Lieb algebras and Kauffman bracket polynomials. These invariants can be understood as a first-order extension of Witten-Reshetikhin-Turaev invariants, which can be reformulated following our approach in the case of rational homology spheres.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2020-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84731137","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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