Pub Date : 2020-08-27DOI: 10.1007/978-3-030-55928-1_5
Philip L. Bowers
{"title":"Combinatorics Encoding Geometry: The Legacy of Bill Thurston in the Story of One Theorem","authors":"Philip L. Bowers","doi":"10.1007/978-3-030-55928-1_5","DOIUrl":"https://doi.org/10.1007/978-3-030-55928-1_5","url":null,"abstract":"","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"47 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79015290","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-08-14DOI: 10.1142/S0218216521500097
C. Adams, Nikhil Agarwal, R. Allen, Tirasan Khandhawit, Alex Simons, Rebecca R. Winarski, Mary Wootters
We improve the upper bound on the superbridge index $sb[K]$ of a knot type $[K]$ in terms of the bridge index $b[K]$ from $sb[K] leq 5b -3$ to $sb[K]leq 3b[k] - 1$.
{"title":"Superbridge and bridge indices for knots","authors":"C. Adams, Nikhil Agarwal, R. Allen, Tirasan Khandhawit, Alex Simons, Rebecca R. Winarski, Mary Wootters","doi":"10.1142/S0218216521500097","DOIUrl":"https://doi.org/10.1142/S0218216521500097","url":null,"abstract":"We improve the upper bound on the superbridge index $sb[K]$ of a knot type $[K]$ in terms of the bridge index $b[K]$ from $sb[K] leq 5b -3$ to $sb[K]leq 3b[k] - 1$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89954729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we introduce a class of local representations of the braid group derived from a meld of the Burau representation and the Rittenberg representations, here thus called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t in k$ the loop-Hecke parameter. We prove the following: $LH_n$ is finite dimensional over a field. The natural inclusion $LB_n rightarrow LB_{n+1}$ passes to an inclusion $SP_n rightarrow SP_{n+1}$. Over $k=mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. item For $t^2 neq 1$ then $LH_n cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.
我们引入了由环辫群$LB_n$通知的普通Hecke代数的一个推广$LH_n$及其Burau表示的推广。普通赫克代数具有许多显著的算术性质和表示理论性质，具有广泛的应用。我们发现$LH_n$具有这些性质中的几个类似物。特别地，我们引入了一类由Burau表示和Rittenberg表示融合而来的辫群的局部表示，在这里称为Burau-Rittenberg表示。在其最超对称的情况下，有些神秘的异常抵消发生了，因此，布劳-里腾堡表示延伸到一个循环布劳-里腾堡表示。这个因子通过$LH_n$。设$SP_n$表示相应的商代数，$k$表示接地环，$t in k$表示loop-Hecke参数。我们证明了$LH_n$在一个域上是有限维的。自然包含$LB_n rightarrow LB_{n+1}$传递给包含$SP_n rightarrow SP_{n+1}$。在$k=mathbb{C}$上，$SP_n / rad $一般是由Pascal三角形给出的维数(和Bratteli图)的简单矩阵代数的和。我们确定了$SP_n$表示理论的其他基本不变量:Cartan分解矩阵;还有箭袋，是a型的。除了$t= 1$外，$SP_n $的结构与$t$参数无关。 item 对于$t^2 neq 1$，那么$LH_n cong SP_n$至少到$n=7$(对于$t=-1$，它们不是同构的$n>2$;对于$t=1$，它们不是同构的(对于$n>1$)。最后，我们讨论了由这种构造在拓扑学、表示理论和组合学中引起的一些其他有趣的问题。
{"title":"Generalisations of Hecke algebras from Loop Braid Groups","authors":"C. Damiani, Paul Martin, E. Rowell","doi":"10.2140/pjm.2023.323.31","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.31","url":null,"abstract":"We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we introduce a class of local representations of the braid group derived from a meld of the Burau representation and the Rittenberg representations, here thus called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t in k$ the loop-Hecke parameter. We prove the following: \u0000$LH_n$ is finite dimensional over a field. \u0000The natural inclusion $LB_n rightarrow LB_{n+1}$ passes to an inclusion $SP_n rightarrow SP_{n+1}$. \u0000Over $k=mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. \u0000We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. \u0000The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. item For $t^2 neq 1$ then $LH_n cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). \u0000Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"2 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76036149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $min mathbb{Z}_{>0}$, denote by $mathscr{T}(kappa,m)$ (resp. $mathscr{Q}(kappa,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $kappa$ is divisible by $6$ (resp. by $4$), then $mathscr{T}(kappa,m)sim c_3(kappa)m^{2g+|kappa|-2}$ (resp. $mathscr{Q}(kappa,m) sim c_4(kappa)m^{2g+|kappa|-2}$), where $c_3(kappa) in mathbb{Q}cdot(sqrt{3}pi)^{2g+|kappa|-2}$ and $c_4(kappa)in mathbb{Q}cdotpi^{2g+|kappa|-2}$. The key ingredient of the proof is a result of J. Kollar on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.
{"title":"Variation of Hodge structure and enumerating tilings of surfaces by triangles and squares","authors":"Vincent Koziarz, Duc-Manh Nguyen","doi":"10.5802/JEP.159","DOIUrl":"https://doi.org/10.5802/JEP.159","url":null,"abstract":"Let $S$ be a connected closed oriented surface of genus $g$. Given a triangulation (resp. quadrangulation) of $S$, define the index of each of its vertices to be the number of edges originating from this vertex minus $6$ (resp. minus $4$). Call the set of integers recording the non-zero indices the profile of the triangulation (resp. quadrangulation). If $kappa$ is a profile for triangulations (resp. quadrangulations) of $S$, for any $min mathbb{Z}_{>0}$, denote by $mathscr{T}(kappa,m)$ (resp. $mathscr{Q}(kappa,m)$) the set of (equivalence classes of) triangulations (resp. quadrangulations) with profile $kappa$ which contain at most $m$ triangles (resp. squares). In this paper, we will show that if $kappa$ is a profile for triangulations (resp. for quadrangulations) of $S$ such that none of the indices in $kappa$ is divisible by $6$ (resp. by $4$), then $mathscr{T}(kappa,m)sim c_3(kappa)m^{2g+|kappa|-2}$ (resp. $mathscr{Q}(kappa,m) sim c_4(kappa)m^{2g+|kappa|-2}$), where $c_3(kappa) in mathbb{Q}cdot(sqrt{3}pi)^{2g+|kappa|-2}$ and $c_4(kappa)in mathbb{Q}cdotpi^{2g+|kappa|-2}$. The key ingredient of the proof is a result of J. Kollar on the link between the curvature of the Hogde metric on vector subbundles of a variation of Hodge structure over algebraic varieties, and Chern classes of their extensions. By the same method, we also obtain the rationality (up to some power of $pi$) of the Masur-Veech volume of arithmetic affine submanifolds of translation surfaces that are transverse to the kernel foliation.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"169 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75651003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-08DOI: 10.2140/agt.2023.23.1097
Fan Ye
This paper studies a special family of (1,1) knots called constrained knots, which includes 2-bridge knots and simple knots. They are parameterized by five parameters and characterized by the distribution of spin^c structures of intersection points in (1,1) diagrams. Their knot Floer homologies are calculated and the complete classification is obtained. Some examples of constrained knots come from links related to 2-bridge knots and 1-bridge braids. As an application, Heegaard Floer theory is studied for orientable 1-cusped hyperbolic manifolds that have ideal triangulations with at most 5 ideal tetrahedra.
{"title":"Constrained knots in lens spaces","authors":"Fan Ye","doi":"10.2140/agt.2023.23.1097","DOIUrl":"https://doi.org/10.2140/agt.2023.23.1097","url":null,"abstract":"This paper studies a special family of (1,1) knots called constrained knots, which includes 2-bridge knots and simple knots. They are parameterized by five parameters and characterized by the distribution of spin^c structures of intersection points in (1,1) diagrams. Their knot Floer homologies are calculated and the complete classification is obtained. Some examples of constrained knots come from links related to 2-bridge knots and 1-bridge braids. As an application, Heegaard Floer theory is studied for orientable 1-cusped hyperbolic manifolds that have ideal triangulations with at most 5 ideal tetrahedra.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"34 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72579669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-02DOI: 10.1142/S0218216521500152
Kanako Oshiro, Natsumi Oyamaguchi
In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings of spatial graph diagrams.
{"title":"Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions","authors":"Kanako Oshiro, Natsumi Oyamaguchi","doi":"10.1142/S0218216521500152","DOIUrl":"https://doi.org/10.1142/S0218216521500152","url":null,"abstract":"In this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knot-theoretic ternary-quasigroups and region colorings of spatial graph diagrams.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87476039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.
{"title":"Cyclic branched covers of alternating knots","authors":"L. Paoluzzi","doi":"10.5802/AHL.89","DOIUrl":"https://doi.org/10.5802/AHL.89","url":null,"abstract":"For any integer n > 2, the n-fold cyclic branched cover M of an alternating prime knot K in the 3-sphere determines K, meaning that if K is a knot in the 3-sphere that is not equivalent to K then its n-fold cyclic branched cover cannot be homeomorphic to M.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"189 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79448757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.
{"title":"Reeb Spaces of Smooth Functions on Manifolds","authors":"O. Saeki","doi":"10.1093/IMRN/RNAA301","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA301","url":null,"abstract":"The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81785534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the Liechti-Strenner's example for the closed nonorientable surface in cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial $p(x)$ of the action induced on the first cohomology nonpositive or all but the first coefficient of $p(x) (x pm 1)^2$, $p(x) (x^2 pm 1)$, or $p(x) (x^2 pm x + 1)$ nonpositive.
{"title":"Remarks on the Liechti-Strenner's examples having small dilatations","authors":"J. Ham, Joongul Lee","doi":"10.4134/CKMS.C190365","DOIUrl":"https://doi.org/10.4134/CKMS.C190365","url":null,"abstract":"We show that the Liechti-Strenner's example for the closed nonorientable surface in cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial of the action induced on the first cohomology nonpositive. We also show that the Liechti-Strenner's example of orientation-reversing homeomorphism for the closed orientable surface in cite{LiechtiStrenner18} minimizes the dilatation within the class of pseudo-Anosov homeomorphisms with an orientable invariant foliation and all but the first coefficient of the characteristic polynomial $p(x)$ of the action induced on the first cohomology nonpositive or all but the first coefficient of $p(x) (x pm 1)^2$, $p(x) (x^2 pm 1)$, or $p(x) (x^2 pm x + 1)$ nonpositive.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73440366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-25DOI: 10.13137/2464-8728/30920
M. Mecchia, Andrea Seppi
It is well known that, among closed spherical Seifert three-manifolds, only lens spaces and prism manifolds admit several Seifert fibrations which are not equivalent up to diffeomorphism. Moreover the former admit infinitely many fibrations, and the latter exactly two. In this work, we analyse the non-uniqueness phenomenon for orbifold Seifert fibrations. For any closed spherical Seifert three-orbifold, we determine the number of its inequivalent fibrations. When these are in a finite number (in fact, at most three) we provide a complete list. In case of infinitely many fibrations, we describe instead an algorithmic procedure to determine whether two closed spherical Seifert orbifolds are diffeomorphic.
{"title":"On the diffeomorphism type of Seifert fibered spherical 3-orbifolds","authors":"M. Mecchia, Andrea Seppi","doi":"10.13137/2464-8728/30920","DOIUrl":"https://doi.org/10.13137/2464-8728/30920","url":null,"abstract":"It is well known that, among closed spherical Seifert three-manifolds, only lens spaces and prism manifolds admit several Seifert fibrations which are not equivalent up to diffeomorphism. Moreover the former admit infinitely many fibrations, and the latter exactly two. In this work, we analyse the non-uniqueness phenomenon for orbifold Seifert fibrations. For any closed spherical Seifert three-orbifold, we determine the number of its inequivalent fibrations. When these are in a finite number (in fact, at most three) we provide a complete list. In case of infinitely many fibrations, we describe instead an algorithmic procedure to determine whether two closed spherical Seifert orbifolds are diffeomorphic.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87507972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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