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Non-loose negative torus knots 非松散的负环面结
IF 1.1 2区 数学
Quantum Topology Pub Date : 2020-01-21 DOI: 10.4171/qt/169
Irena Matkovič
{"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":"96 1","pages":""},"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}
引用次数: 4
Exotic Lagrangian tori in Grassmannians 格拉斯曼语中的奇异拉格朗日环面
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-24 DOI: 10.4171/qt/173
Marco Castronovo
{"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":"46 1","pages":""},"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}
引用次数: 4
Flags and tangles 旗帜和缠结
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-09 DOI: 10.4171/QT/157
F. Haiden
{"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":"15 1","pages":""},"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}
引用次数: 1
Seifert hypersurfaces of 2-knots and Chern–Simons functional 2节的Seifert超曲面和chen - simons泛函
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-10-05 DOI: 10.4171/qt/165
Masaki Taniguchi
{"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":"86 1","pages":""},"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}
引用次数: 4
Link homology theories and ribbon concordances 链接同源理论和条带一致性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-09-16 DOI: 10.4171/qt/162
Sungkyung Kang
{"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":"158 1","pages":""},"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}
引用次数: 9
Arbitrarily large torsion in Khovanov cohomology Khovanov上同调中的任意大扭转
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-09-16 DOI: 10.4171/QT/149
Sujoy Mukherjee, D. Schuetz
{"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":"67 1","pages":""},"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}
引用次数: 4
The Roger–Yang skein algebra and the decorated Teichmüller space 罗杰-杨交织代数和装饰的teichm<e:1>勒空间
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-09-06 DOI: 10.4171/QT/150
Han-Bom Moon, H. Wong
{"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":"80 1","pages":""},"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}
引用次数: 6
Four dimensional topological quantum field theories from $G$-crossed braided categories 从$G$交叉编织范畴出发的四维拓扑量子场论
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-09-06 DOI: 10.4171/qt/128
Shawn X. Cui
{"title":"Four dimensional topological quantum field theories from $G$-crossed braided categories","authors":"Shawn X. Cui","doi":"10.4171/qt/128","DOIUrl":"https://doi.org/10.4171/qt/128","url":null,"abstract":"","PeriodicalId":51331,"journal":{"name":"Quantum Topology","volume":"52 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2019-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89919539","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}
引用次数: 21
HOMFLYPT homology for links in handlebodies via type A Soergel bimodules 通过A型Soergel双模的柄体连杆的HOMFLYPT同源性
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-08-19 DOI: 10.4171/QT/152
David E. V. Rose, D. Tubbenhauer
{"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":"34 1","pages":""},"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}
引用次数: 9
Embedding Deligne's category $mathrm{Underline{Re}p}(S_t)$ in the Heisenberg category 在Heisenberg范畴中嵌入Deligne的范畴$ mathm {Underline{Re}p}(S_t)$
IF 1.1 2区 数学
Quantum Topology Pub Date : 2019-05-14 DOI: 10.4171/QT/147
Samuel Nyobe Likeng, Alistair Savage
{"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":"109 1","pages":""},"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}
引用次数: 0
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