For each plabic graph of type (k,n) in the sense of Postnikov satisfying a smallness condition, we construct a nondisplaceable monotone Lagrangian torus in the complex Grassmannian Gr(k,n). Among these we find examples that bound the same number of families of Maslov 2 pseudoholomorphic disks, whose Hamiltonian isotopy classes are distinguished by the number of critical points in different algebraic torus charts of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. The tori are fibers of local regular Lagrangian fibrations over Okounkov bodies for the frozen anticanonical divisor, which is singled out by the cluster structure of the Grassmannian and has been studied by Rietsch-Williams. Lagrangian tori of plabic graphs related by a combinatorial square move have disk potentials connected by a 3-term Plucker relation, while their Newton polytopes undergo width 2 mutation in the sense of Akhtar-Coates-Galkin-Kasprzyk.
{"title":"Exotic Lagrangian tori in Grassmannians","authors":"Marco Castronovo","doi":"10.4171/qt/173","DOIUrl":"https://doi.org/10.4171/qt/173","url":null,"abstract":"For each plabic graph of type (k,n) in the sense of Postnikov satisfying a smallness condition, we construct a nondisplaceable monotone Lagrangian torus in the complex Grassmannian Gr(k,n). Among these we find examples that bound the same number of families of Maslov 2 pseudoholomorphic disks, whose Hamiltonian isotopy classes are distinguished by the number of critical points in different algebraic torus charts of a mirror Landau-Ginzburg model proposed by Marsh-Rietsch. The tori are fibers of local regular Lagrangian fibrations over Okounkov bodies for the frozen anticanonical divisor, which is singled out by the cluster structure of the Grassmannian and has been studied by Rietsch-Williams. Lagrangian tori of plabic graphs related by a combinatorial square move have disk potentials connected by a 3-term Plucker relation, while their Newton polytopes undergo width 2 mutation in the sense of Akhtar-Coates-Galkin-Kasprzyk.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78181876","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 two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.
{"title":"Flags and tangles","authors":"F. Haiden","doi":"10.4171/QT/157","DOIUrl":"https://doi.org/10.4171/QT/157","url":null,"abstract":"We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88346948","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 introduce a real-valued functional on the $SU(2)$-representation space of the knot group for any oriented $2$-knot. We calculate the functionals for ribbon $2$-knots and the twisted spun $2$-knots of torus knots, $2$-bridge knots and Montesinos knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of $K$. As a corollary, we show that every oriented $2$-knot having a homology $3$-sphere of a certain class as its Seifert hypersurface admits an $SU(2)$-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology $3$-sphere into a negative definite $4$-manifold to $SU(2)$-representations of their fundamental groups. For example, we prove that every closed definite $4$-manifold containing $Sigma(2,3,5,7)$ as a submanifold has an uncountable family of $SU(2)$-representations of its fundamental group. This implies that every $2$-knot having $Sigma(2,3,5,7)$ as a Seifert hypersurface has an uncountable family of $SU(2)$-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.
{"title":"Seifert hypersurfaces of 2-knots and Chern–Simons functional","authors":"Masaki Taniguchi","doi":"10.4171/qt/165","DOIUrl":"https://doi.org/10.4171/qt/165","url":null,"abstract":"We introduce a real-valued functional on the $SU(2)$-representation space of the knot group for any oriented $2$-knot. We calculate the functionals for ribbon $2$-knots and the twisted spun $2$-knots of torus knots, $2$-bridge knots and Montesinos knots. We show several properties of the images of the functionals including a connected sum formula and relationship to the Chern-Simons functionals of Seifert hypersurfaces of $K$. As a corollary, we show that every oriented $2$-knot having a homology $3$-sphere of a certain class as its Seifert hypersurface admits an $SU(2)$-irreducible representation of a knot group. Moreover, we also relate the existence of embeddings from a homology $3$-sphere into a negative definite $4$-manifold to $SU(2)$-representations of their fundamental groups. For example, we prove that every closed definite $4$-manifold containing $Sigma(2,3,5,7)$ as a submanifold has an uncountable family of $SU(2)$-representations of its fundamental group. This implies that every $2$-knot having $Sigma(2,3,5,7)$ as a Seifert hypersurface has an uncountable family of $SU(2)$-representations of its knot group. The proofs of these results use several techniques from instanton Floer theory.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85876495","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}
It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.
{"title":"Link homology theories and ribbon concordances","authors":"Sungkyung Kang","doi":"10.4171/qt/162","DOIUrl":"https://doi.org/10.4171/qt/162","url":null,"abstract":"It was recently proved by several authors that ribbon concordances induce injective maps in knot Floer homology, Khovanov homology, and the Heegaard Floer homology of the branched double cover. We give a simple proof of a similar statement in a more general setting, which includes knot Floer homology, Khovanov-Rozansky homologies, and all conic strong Khovanov-Floer theories. This gives a philosophical answer to the question of which aspects of a link TQFT make it injective under ribbon concordances.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72472912","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":"Arbitrarily large torsion in Khovanov cohomology","authors":"Sujoy Mukherjee, D. Schuetz","doi":"10.4171/QT/149","DOIUrl":"https://doi.org/10.4171/QT/149","url":null,"abstract":"For any positive integer $k$ and $pin {3,5,7}$ we construct a link which has a direct summand $mathbb{Z}/p^kmathbb{Z}$ in its Khovanov cohomology.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89800452","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}
Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Poisson algebra homomorphism is injective, and the skein algebra they defined have no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.
{"title":"The Roger–Yang skein algebra and the decorated Teichmüller space","authors":"Han-Bom Moon, H. Wong","doi":"10.4171/QT/150","DOIUrl":"https://doi.org/10.4171/QT/150","url":null,"abstract":"Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmuller space. In this paper, we consider surfaces with punctures which is not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang's Poisson algebra homomorphism is injective, and the skein algebra they defined have no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74170143","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 define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.
{"title":"HOMFLYPT homology for links in handlebodies via type A Soergel bimodules","authors":"David E. V. Rose, D. Tubbenhauer","doi":"10.4171/QT/152","DOIUrl":"https://doi.org/10.4171/QT/152","url":null,"abstract":"We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid group, and a family of categorical actions built from complexes of (singular) Soergel bimodules.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75305671","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 define a faithful linear monoidal functor from the partition category, and hence from Deligne’s category Rep(St), to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.
{"title":"Embedding Deligne's category $mathrm{Underline{Re}p}(S_t)$ in the Heisenberg category","authors":"Samuel Nyobe Likeng, Alistair Savage","doi":"10.4171/QT/147","DOIUrl":"https://doi.org/10.4171/QT/147","url":null,"abstract":"We define a faithful linear monoidal functor from the partition category, and hence from Deligne’s category Rep(St), to the additive Karoubi envelope of the Heisenberg category. We show that the induced map on Grothendieck rings is injective and corresponds to the Kronecker coproduct on symmetric functions.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80843270","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 physical 3d $mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $hat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $hat{Z}_a(q)$ for some hyperbolic 3-manifolds.
{"title":"A two-variable series for knot complements","authors":"S. Gukov, Ciprian Manolescu","doi":"10.4171/QT/145","DOIUrl":"https://doi.org/10.4171/QT/145","url":null,"abstract":"The physical 3d $mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk. Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial. The terms in this series should satisfy a recurrence given by the quantum A-polynomial. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants $hat{Z}_{a}(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand $hat{Z}_a(q)$ for some hyperbolic 3-manifolds.","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":null,"pages":null},"PeriodicalIF":1.1,"publicationDate":"2019-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79788928","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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