We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $nu^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.
{"title":"Genus One Cobordisms Between Torus Knots","authors":"P. Feller, Junghwan Park","doi":"10.1093/IMRN/RNAA027","DOIUrl":"https://doi.org/10.1093/IMRN/RNAA027","url":null,"abstract":"We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $nu^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"8 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82253957","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 consider harmonic measures that arise from a finitely supported random walk on the mapping class group whose support generates a non-elementary subgroup. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.
{"title":"Statistical Hyperbolicity for Harmonic Measure","authors":"Vaibhav Gadre, Luke Jeffreys","doi":"10.1093/imrn/rnaa277","DOIUrl":"https://doi.org/10.1093/imrn/rnaa277","url":null,"abstract":"We consider harmonic measures that arise from a finitely supported random walk on the mapping class group whose support generates a non-elementary subgroup. We prove that Teichmuller space with the Teichmuller metric is statistically hyperbolic for such a harmonic measure.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"107 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77814406","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 : 2019-09-30DOI: 10.1215/00192082-8642515
Nozomu Sekino
We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev density theorem.
{"title":"Lens spaces which are realizable as closures of homology cobordisms over planar surfaces","authors":"Nozomu Sekino","doi":"10.1215/00192082-8642515","DOIUrl":"https://doi.org/10.1215/00192082-8642515","url":null,"abstract":"We determine the condition on a given lens space having a realization as a closure of homology cobordism over a planar surface with a given number of boundary components. As a corollary, we see that every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components. In the proof of this corollary, we use Chebotarev density theorem.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"5 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72631365","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 : 2019-09-20DOI: 10.1142/S0218216521500449
G. Bellettini, M. Paolini, Yi-Sheng Wang
In this paper we use diagrams in categories to construct a complete invariant, the fundamental tree, for closed surfaces in the (based) $3$-sphere, which generalizes the knot group and its peripheral system. From the fundamental tree, we derive some computable invariants that are capable to distinguish inequivalent handlebody links with homeomorphic complements. To prove the completeness of the fundamental tree, we generalize the Kneser conjecture to $3$-manifolds with boundary, a topic interesting in its own right.
{"title":"A complete invariant for closed surfaces in the three-sphere","authors":"G. Bellettini, M. Paolini, Yi-Sheng Wang","doi":"10.1142/S0218216521500449","DOIUrl":"https://doi.org/10.1142/S0218216521500449","url":null,"abstract":"In this paper we use diagrams in categories to construct a complete invariant, the fundamental tree, for closed surfaces in the (based) $3$-sphere, which generalizes the knot group and its peripheral system. From the fundamental tree, we derive some computable invariants that are capable to distinguish inequivalent handlebody links with homeomorphic complements. To prove the completeness of the fundamental tree, we generalize the Kneser conjecture to $3$-manifolds with boundary, a topic interesting in its own right.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73339286","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 : 2019-09-19DOI: 10.2140/agt.2020.20.3127
Emily Shinkle
For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak--McCarthy.
{"title":"Finite rigid sets in arc complexes","authors":"Emily Shinkle","doi":"10.2140/agt.2020.20.3127","DOIUrl":"https://doi.org/10.2140/agt.2020.20.3127","url":null,"abstract":"For any compact, connected, orientable, finite-type surface with marked points other than the sphere with three marked points, we construct a finite rigid set of its arc complex: a finite simplicial subcomplex of its arc complex such that any locally injective map of this set into the arc complex of another surface with arc complex of the same or lower dimension is induced by a homeomorphism of the surfaces, unique up to isotopy in most cases. It follows that if the arc complexes of two surfaces are isomorphic, the surfaces are homeomorphic. We also give an exhaustion of the arc complex by finite rigid sets. This extends the results of Irmak--McCarthy.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"704 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76894611","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}
Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmuller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.
{"title":"Quadratic differentials and circle patterns on complex projective tori","authors":"Wai Yeung Lam","doi":"10.2140/gt.2021.25.961","DOIUrl":"https://doi.org/10.2140/gt.2021.25.961","url":null,"abstract":"Given a triangulation of a closed surface, we consider a cross ratio system that assigns a complex number to every edge satisfying certain polynomial equations per vertex. Every cross ratio system induces a complex projective structure together with a circle pattern on the closed surface. In particular, there is an associated conformal structure. We show that for any triangulated torus, the projection from the space of cross ratio systems with prescribed Delaunay angles to the Teichmuller space is a covering map with at most one branch point. Our approach is based on a notion of discrete holomorphic quadratic differentials.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82414916","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}
This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman--Turaev Lie bialgebra, and the work of various authors on cohomological methods for determining the stable image of generalized Johnson homomorphisms. Various open problems and conjectures are included.
{"title":"Johnson homomorphisms","authors":"R. Hain","doi":"10.4171/emss/36","DOIUrl":"https://doi.org/10.4171/emss/36","url":null,"abstract":"This paper surveys work on generalized Johnson homomorphisms and tools for studying them. The goal is to unite several related threads in the literature and to clarify existing results and relationships among them using Hodge theory. We survey the work of Alekseev, Kawazumi, Kuno and Naef on the Goldman--Turaev Lie bialgebra, and the work of various authors on cohomological methods for determining the stable image of generalized Johnson homomorphisms. Various open problems and conjectures are included.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73776862","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 : 2019-08-22DOI: 10.1215/00127094-2021-0054
V. Delecroix, É. Goujard, P. Zograf, A. Zorich
We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.
{"title":"Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves","authors":"V. Delecroix, É. Goujard, P. Zograf, A. Zorich","doi":"10.1215/00127094-2021-0054","DOIUrl":"https://doi.org/10.1215/00127094-2021-0054","url":null,"abstract":"We express the Masur-Veech volume and the area Siegel-Veech constant of the moduli space of meromorphic quadratic differential with simple poles as polynomials in the intersection numbers of psi-classes supported on the boundary cycles of the Deligne-Mumford compactification of the moduli space of curves. Our formulae are derived from lattice point count involving the Kontsevich volume polynomials that also appear in Mirzakhani's recursion for the Weil-Petersson volumes of the moduli space of bordered hyperbolic Riemann surfaces. A similar formula for the Masur-Veech volume (though without explicit evaluation) was obtained earlier by Mirzakhani through completely different approach. We prove further result: up to an explicit normalization factor depending only on the genus and on the number of cusps, the density of the orbit of any simple closed multicurve computed by Mirzakhani coincides with the density of square-tiled surfaces having horizontal cylinder decomposition associated to the simple closed multicurve. We study the resulting densities in more detail in the special case when there are no cusps. In particular, we compute explicitly the asymptotic frequencies of separating and non-separating simple closed geodesics on a closed hyperbolic surface of genus g for all small genera g and we show that in large genera the separating closed geodesics are exponentially less frequent. We conclude with detailed conjectural description of combinatorial geometry of a random simple closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. This description is conditional to the conjectural asymptotic formula for the Masur-Veech volume in large genera and to the conjectural uniform asymptotic formula for certain sums of intersection numbers of psi-classes in large genera.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76771997","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 : 2019-08-18DOI: 10.1142/s021821652050039x
Benjamin Bode
We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $mathbb{B}_n$ on the set of $n$-adic integers $mathbb{Z}_n$ for all natural numbers $ngeq 2$. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show that an infinite family of braids close to real algebraic links, i.e., links of isolated singularities of real polynomials $mathbb{R}^4tomathbb{R}^2$.
{"title":"Real algebraic links in S3 and braid group actions on the set of n-adic integers","authors":"Benjamin Bode","doi":"10.1142/s021821652050039x","DOIUrl":"https://doi.org/10.1142/s021821652050039x","url":null,"abstract":"We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $mathbb{B}_n$ on the set of $n$-adic integers $mathbb{Z}_n$ for all natural numbers $ngeq 2$. We study some of the properties of these actions such as continuity and transitivity. The construction of the actions involves a new way of associating to any braid $B$ an infinite sequence of braids, whose braid types are invariants of $B$. We present computations for the cases of $n=2$ and $n=3$ and use these to show that an infinite family of braids close to real algebraic links, i.e., links of isolated singularities of real polynomials $mathbb{R}^4tomathbb{R}^2$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87444798","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 : 2019-08-15DOI: 10.1142/s0218216520420080
A. Gill, M. Prabhakar, Andrei Vesnin
Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that that the constructed virtual knots have the same affine index polynomial.
{"title":"Gordian complexes of knots and virtual knots given by region crossing changes and arc shift moves","authors":"A. Gill, M. Prabhakar, Andrei Vesnin","doi":"10.1142/s0218216520420080","DOIUrl":"https://doi.org/10.1142/s0218216520420080","url":null,"abstract":"Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in $mathbb{S}^3$. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using $v$-move and forbidden moves. In this paper we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high dimensional simplex in both the Gordian complexes, i.e., by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that that the constructed virtual knots have the same affine index polynomial.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77199547","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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