Pub Date : 2020-02-24DOI: 10.1142/s0219199720500662
E. Horvat
The lens space $L_{p,q}$ is the orbit space of a $mathbb{Z}_{p}$-action on the three sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in $L_{p,q}$. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in lens spaces, whose lift in the $3$-sphere is a torus link.
{"title":"Algebraic links in lens spaces","authors":"E. Horvat","doi":"10.1142/s0219199720500662","DOIUrl":"https://doi.org/10.1142/s0219199720500662","url":null,"abstract":"The lens space $L_{p,q}$ is the orbit space of a $mathbb{Z}_{p}$-action on the three sphere. We investigate polynomials of two complex variables that are invariant under this action, and thus define links in $L_{p,q}$. We study properties of these links, and their relationship with the classical algebraic links. We prove that all algebraic links in lens spaces are fibered, and obtain results about their Seifert genus. We find some examples of algebraic knots in lens spaces, whose lift in the $3$-sphere is a torus link.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85010472","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 notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $beta$ defining a foliation on a branched simple polyhedron $P$ satisfies $dbeta>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $beta$ on $P$ with $dbeta>0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $beta$ with $dbeta>0$ on the abalone.
R. Benedetti和C. Petronio在研究3流形上接触结构的特征叶形时，引入了分支简单多面体上叶形的s稳定性概念。我们还假设在分支简单多面体$P$上定义一个叶理的1-形式$beta$满足$dbeta>0$，这意味着该叶理是一个接触形式的特征叶理，其Reeb流横向于$P$。在本文中，我们证明了如果$P$上存在一个1-form $beta$且$dbeta>0$，那么我们就能找到一个具有相同性质的1-form $beta$并且是s稳定的。然后，我们证明了正或负流脊上s稳定叶理的简单切点数至少为2，并给出了在鲍鱼上构造$dbeta>0$的1-形$beta$特征叶理的方法。
{"title":"S-stable foliations on flow-spines with transverse Reeb\u0000 flow","authors":"Shin Handa, M. Ishikawa","doi":"10.32917/H2020026","DOIUrl":"https://doi.org/10.32917/H2020026","url":null,"abstract":"The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $beta$ defining a foliation on a branched simple polyhedron $P$ satisfies $dbeta>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $beta$ on $P$ with $dbeta>0$ then we can find a 1-form with the same property and additionally being S-stable. We then prove that the number of simple tangency points of an S-stable foliation on a positive or negative flow-spine is at least 2 and give a recipe for constructing a characteristic foliation of a 1-form $beta$ with $dbeta>0$ on the abalone.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74835446","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}
Author(s): Gorsky, Eugene; Hogancamp, Matthew; Wedrich, Paul | Abstract: We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus.
{"title":"Derived Traces of Soergel Categories","authors":"E. Gorsky, Matthew Hogancamp, Paul Wedrich","doi":"10.1093/IMRN/RNAB019","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB019","url":null,"abstract":"Author(s): Gorsky, Eugene; Hogancamp, Matthew; Wedrich, Paul | Abstract: We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87749954","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 : 2020-02-03DOI: 10.1007/978-3-030-55928-1_3
M. Sakuma
{"title":"A Survey of the Impact of Thurston’s Work on Knot Theory","authors":"M. Sakuma","doi":"10.1007/978-3-030-55928-1_3","DOIUrl":"https://doi.org/10.1007/978-3-030-55928-1_3","url":null,"abstract":"","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75508250","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}
Hirotaka Akiyoshi, Ken'ichi Ohshika, J. Parker, M. Sakuma, H. Yoshida
We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.
我们充分证明了Agol关于由两个抛物变换生成的非自由Kleinian群的分类的公告。
{"title":"Classification of non-free Kleinian groups generated by two parabolic transformations","authors":"Hirotaka Akiyoshi, Ken'ichi Ohshika, J. Parker, M. Sakuma, H. Yoshida","doi":"10.1090/TRAN/8246","DOIUrl":"https://doi.org/10.1090/TRAN/8246","url":null,"abstract":"We give a full proof to Agol's announcement on the classification of non-free Kleinian groups generated by two parabolic transformations.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":" 41","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91515729","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, under weak assumptions, the automorphism group of a ${rm CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.
{"title":"Automorphisms of Contact Graphs of CAT(0) Cube Complexes","authors":"Elia Fioravanti","doi":"10.1093/imrn/rnaa280","DOIUrl":"https://doi.org/10.1093/imrn/rnaa280","url":null,"abstract":"We show that, under weak assumptions, the automorphism group of a ${rm CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen's contact graph $mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov's theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim-Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated to Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well-behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82425472","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 prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.
我们证明了伪ananosov映射类在泛型的某些概念上是泛型的，这反映了我们正在处理映射类。
{"title":"Genericity of pseudo-Anosov mapping classes, when seen as mapping classes","authors":"V. Erlandsson, J. Souto, Jing Tao","doi":"10.4171/LEM/66-3/4-6","DOIUrl":"https://doi.org/10.4171/LEM/66-3/4-6","url":null,"abstract":"We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83381514","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 : 2019-12-28DOI: 10.1142/s0218216520500844
Karma Istanbouli, Sam Nelson
We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.
{"title":"Quandle Module Quivers","authors":"Karma Istanbouli, Sam Nelson","doi":"10.1142/s0218216520500844","DOIUrl":"https://doi.org/10.1142/s0218216520500844","url":null,"abstract":"We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant with specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same quandle module polynomial.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"88 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76400343","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 : 2019-12-13DOI: 10.2140/involve.2020.13.633
Andrew Ducharme, E. Peters
We explore free knot diagrams, which are projections of knots into the plane which don't record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple families of free knots are completely worked out. We make some conjectures (supported by computer-generated data) about bounds on the probability of a knot arising from a fixed free diagram being the unknot, or being the trefoil.
{"title":"Combinatorial random knots","authors":"Andrew Ducharme, E. Peters","doi":"10.2140/involve.2020.13.633","DOIUrl":"https://doi.org/10.2140/involve.2020.13.633","url":null,"abstract":"We explore free knot diagrams, which are projections of knots into the plane which don't record over/under data at crossings. We consider the combinatorial question of which free knot diagrams give which knots and with what probability. Every free knot diagram is proven to produce trefoil knots, and certain simple families of free knots are completely worked out. We make some conjectures (supported by computer-generated data) about bounds on the probability of a knot arising from a fixed free diagram being the unknot, or being the trefoil.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82396592","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 provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety of the fundamental group, which for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every $SU(2)$ representation has cyclic image.
{"title":"Surgery obstructions and character varieties","authors":"Steven Sivek, Raphael Zentner","doi":"10.1090/tran/8596","DOIUrl":"https://doi.org/10.1090/tran/8596","url":null,"abstract":"We provide infinitely many rational homology 3-spheres with weight-one fundamental groups which do not arise from Dehn surgery on knots in $S^3$. In contrast with previously known examples, our proofs do not require any gauge theory or Floer homology. Instead, we make use of the $SU(2)$ character variety of the fundamental group, which for these manifolds is particularly simple: they are all $SU(2)$-cyclic, meaning that every $SU(2)$ representation has cyclic image.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"127 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88698991","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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