Pub Date : 2021-04-23DOI: 10.11606/T.45.2021.TDE-11052021-020459
Arcelino Bruno Lobato do Nascimento
Thurston obtained a combinatorial characterization for generic branched self-coverings that preserve the orientation of the oriented 2-sphere by associating a planar graph to them [arXiv:1502.04760]. In this work, the Thurston result is generalized to any branched covering of the oriented 2-sphere. To achieve that the notion of local balance introduced by Thurston is generalized. As an application, a new proof for a Theorem of Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko [MR1888795], [MR2552110] is obtained. This theorem corresponded to a special case of the B. & M. Shapiro conjecture. In this case, it refers to generic rational functions stating that a generic rational function $ R : mathbb{C}mathbb{P}^1 rightarrow mathbb{C}mathbb{P}^1$ with only real critical points can be transformed by post-composition with an automorphism of $mathbb{C}mathbb{P}^1$ into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.
{"title":"Branched coverings of the 2-sphere","authors":"Arcelino Bruno Lobato do Nascimento","doi":"10.11606/T.45.2021.TDE-11052021-020459","DOIUrl":"https://doi.org/10.11606/T.45.2021.TDE-11052021-020459","url":null,"abstract":"Thurston obtained a combinatorial characterization for generic branched self-coverings that preserve the orientation of the oriented 2-sphere by associating a planar graph to them [arXiv:1502.04760]. In this work, the Thurston result is generalized to any branched covering of the oriented 2-sphere. To achieve that the notion of local balance introduced by Thurston is generalized. As an application, a new proof for a Theorem of Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko [MR1888795], [MR2552110] is obtained. This theorem corresponded to a special case of the B. & M. Shapiro conjecture. In this case, it refers to generic rational functions stating that a generic rational function $ R : mathbb{C}mathbb{P}^1 rightarrow mathbb{C}mathbb{P}^1$ with only real critical points can be transformed by post-composition with an automorphism of $mathbb{C}mathbb{P}^1$ into a quotient of polynomials with real coefficients. Operations against balanced graphs are introduced.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73787324","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}
Fock and Goncharov introduced cluster ensembles, providing a framework for coordinates on varieties of surface representations into Lie groups, as well as a complete construction for groups of type $A_n$. Later, Zickert, Le, and Ip described, using differing methods, how to apply this framework for other Lie group types. Zickert also showed that this framework applies to triangulated $3$-manifolds. We present a complete, general construction, based on work of Fomin and Zelevinsky. In particular, we complete the picture for the remaining cases: Lie groups of types $F_4$, $E_6$, $E_7$, and $E_8$.
{"title":"Fock–Goncharov coordinates for semisimple Lie groups","authors":"S. Gilles","doi":"10.13016/2IHM-X8LS","DOIUrl":"https://doi.org/10.13016/2IHM-X8LS","url":null,"abstract":"Fock and Goncharov introduced cluster ensembles, providing a framework for coordinates on varieties of surface representations into Lie groups, as well as a complete construction for groups of type $A_n$. Later, Zickert, Le, and Ip described, using differing methods, how to apply this framework for other Lie group types. Zickert also showed that this framework applies to triangulated $3$-manifolds. We present a complete, general construction, based on work of Fomin and Zelevinsky. In particular, we complete the picture for the remaining cases: Lie groups of types $F_4$, $E_6$, $E_7$, and $E_8$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75478445","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 : 2020-12-14DOI: 10.1142/S1793525321500394
Adalet Cengel, Mustafa Korkmaz
For $ggeq 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes all Lefschetz fibrations are not realized as slopes.
{"title":"Low-Slope Lefschetz Fibrations","authors":"Adalet Cengel, Mustafa Korkmaz","doi":"10.1142/S1793525321500394","DOIUrl":"https://doi.org/10.1142/S1793525321500394","url":null,"abstract":"For $ggeq 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes all Lefschetz fibrations are not realized as slopes.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80597173","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 : 2020-12-11DOI: 10.1016/J.TOPOL.2021.107837
Nozomu Sekino
{"title":"The existence of homologically fibered links and solutions of some equations.","authors":"Nozomu Sekino","doi":"10.1016/J.TOPOL.2021.107837","DOIUrl":"https://doi.org/10.1016/J.TOPOL.2021.107837","url":null,"abstract":"","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"6 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91431966","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 $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $Mod(M_n)$ is an extension of $Out(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $Mod(M_n)$ is the semidirect product of $Out(F_n)$ by $(mathbb{Z}/2)^n$, which $Out(F_n)$ acts on via the dual of the natural surjection $Out(F_n) rightarrow GL_n(mathbb{Z}/2)$. Our splitting takes $Out(F_n)$ to the subgroup of $Mod(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbach's original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.
{"title":"The mapping class group of connect sums of $S^2 times S^1$","authors":"Tara E. Brendle, N. Broaddus, Andrew Putman","doi":"10.1090/tran/8758","DOIUrl":"https://doi.org/10.1090/tran/8758","url":null,"abstract":"Let $M_n$ be the connect sum of $n$ copies of $S^2 times S^1$. A classical theorem of Laudenbach says that the mapping class group $Mod(M_n)$ is an extension of $Out(F_n)$ by a group $(mathbb{Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $Mod(M_n)$ is the semidirect product of $Out(F_n)$ by $(mathbb{Z}/2)^n$, which $Out(F_n)$ acts on via the dual of the natural surjection $Out(F_n) rightarrow GL_n(mathbb{Z}/2)$. Our splitting takes $Out(F_n)$ to the subgroup of $Mod(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbach's original proof, including the identification of the twist subgroup with $(mathbb{Z}/2)^n$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"91 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85766892","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 cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 le d(K,K^r) le 2g_4(K) and it is known that there are knots K for which d(K,K^r) is arbitrarily large. Here it is shown that for any knot for which g_4(K) = g_3(K) (such as non-slice knots with g_3(K) = 1 or strongly quasi-positive knots), one has that d(K,K^r) is strictly less that twice g_4(K). It is shown that for arbitrary positive g, there exist knots for which d(K,K^r) = g = g_4(K). There are no known examples for which d(K,K^r) > g_4(K).
结点之间的协距d(K,J)等于四格g_4(k# -J)。我们考虑d(K,K^r)其中K^r是K的倒数，它是初等的0 le d(K,K^r) le 2g_4(K)我们知道有一些结点K d(K,K^r)是任意大的。本文证明了对于任意g_4(K) = g_3(K)的结(如g_3(K) = 1的非切片结或强拟正结)，d(K,K^r)严格小于2倍g_4(K)。证明了对于任意正g，存在d(K,K^r) = g = g_4(K)的结点。没有已知的d(K,K^r) > g_4(K)的例子。
{"title":"The cobordism distance between a knot and its reverse","authors":"C. Livingston","doi":"10.1090/proc/15809","DOIUrl":"https://doi.org/10.1090/proc/15809","url":null,"abstract":"The cobordism distance between knots, d(K,J), equals the four-genus g_4(K # -J). We consider d(K,K^r), where K^r is the reverse of K. It is elementary that 0 le d(K,K^r) le 2g_4(K) and it is known that there are knots K for which d(K,K^r) is arbitrarily large. Here it is shown that for any knot for which g_4(K) = g_3(K) (such as non-slice knots with g_3(K) = 1 or strongly quasi-positive knots), one has that d(K,K^r) is strictly less that twice g_4(K). It is shown that for arbitrary positive g, there exist knots for which d(K,K^r) = g = g_4(K). There are no known examples for which d(K,K^r) > g_4(K).","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74666611","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 : 2020-11-10DOI: 10.15673/tmgc.v13i4.1855
I. Dynnikov, M. Prasolov
In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.
{"title":"Rectangular diagrams of surfaces: the basic moves","authors":"I. Dynnikov, M. Prasolov","doi":"10.15673/tmgc.v13i4.1855","DOIUrl":"https://doi.org/10.15673/tmgc.v13i4.1855","url":null,"abstract":"In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to prove a Reidemeister type theorem for rectangular diagrams of surfaces.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"21 2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77480989","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 space X we study the topology of the space of embeddings of X into $mathbb{R}^d$ through the combinatorics of triangulations of X. We give a simple combinatorial formula for upper bounds for the largest dimension of a sphere that antipodally maps into the space of embeddings. This result summarizes and extends results about the nonembeddability of complexes into $mathbb{R}^d$, the nonexistence of nonsingular bilinear maps, and the study of embeddings into $mathbb{R}^d$ up to isotopy, such as the chirality of spatial graphs.
{"title":"Spaces of embeddings: Nonsingular bilinear maps, chirality, and their generalizations","authors":"F. Frick, Michael C. Harrison","doi":"10.1090/proc/15752","DOIUrl":"https://doi.org/10.1090/proc/15752","url":null,"abstract":"Given a space X we study the topology of the space of embeddings of X into $mathbb{R}^d$ through the combinatorics of triangulations of X. We give a simple combinatorial formula for upper bounds for the largest dimension of a sphere that antipodally maps into the space of embeddings. This result summarizes and extends results about the nonembeddability of complexes into $mathbb{R}^d$, the nonexistence of nonsingular bilinear maps, and the study of embeddings into $mathbb{R}^d$ up to isotopy, such as the chirality of spatial graphs.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"09 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89715209","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 : 2020-10-21DOI: 10.13137/2464-8728/30958
A. Egorov, A. Vesnin
By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.
{"title":"Volume estimates for right-angled hyperbolic polyhedra","authors":"A. Egorov, A. Vesnin","doi":"10.13137/2464-8728/30958","DOIUrl":"https://doi.org/10.13137/2464-8728/30958","url":null,"abstract":"By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space $mathbb H^{3}$ are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact right-angled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"199 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75533498","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 continue the study of freezing sets for digital images introduced in [4, 2, 3]. We prove methods for obtaining freezing sets for digital images (X, c_i) for X subset Z^2 and i in {1, 2}. We give examples to show how these methods can lead to the determination of minimal freezing sets.
{"title":"Subsets and Freezing Sets in the Digital Plane","authors":"L. Boxer","doi":"10.15672/HUJMS.827556","DOIUrl":"https://doi.org/10.15672/HUJMS.827556","url":null,"abstract":"We continue the study of freezing sets for digital images introduced in [4, 2, 3]. We prove methods for obtaining freezing sets for digital images (X, c_i) for X subset Z^2 and i in {1, 2}. We give examples to show how these methods can lead to the determination of minimal freezing sets.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75869592","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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