We continue the study of freezing sets in digital topology, introduced in [2]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that X is convex.
{"title":"Convexity and freezing sets in digital topology","authors":"L. Boxer","doi":"10.4995/AGT.2021.14185","DOIUrl":"https://doi.org/10.4995/AGT.2021.14185","url":null,"abstract":"We continue the study of freezing sets in digital topology, introduced in [2]. We show how to find a minimal freezing set for a \"thick\" convex disk X in the digital plane Z^2. We give examples showing the significance of the assumption that X is convex.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91109479","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 : 2020-05-11DOI: 10.31390/gradschool_dissertations.5017
Ryan Leigon, Federico Salmoiraghi
We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description. In particular, we establish a conjecture of Zarev, that his gluing map on sutured Floer homology is equivalent to the contact gluing map.
{"title":"Equivalence of Contact Gluing Maps in Sutured Floer Homology","authors":"Ryan Leigon, Federico Salmoiraghi","doi":"10.31390/gradschool_dissertations.5017","DOIUrl":"https://doi.org/10.31390/gradschool_dissertations.5017","url":null,"abstract":"We show that the contact gluing map of Honda, Kazez, and Matic has a natural algebraic description. In particular, we establish a conjecture of Zarev, that his gluing map on sutured Floer homology is equivalent to the contact gluing map.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88875157","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 announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations.
{"title":"The real spectrum compactification of character varieties: characterizations and applications","authors":"M. Burger, A. Iozzi, A. Parreau, M. B. Pozzetti","doi":"10.5802/crmath.123","DOIUrl":"https://doi.org/10.5802/crmath.123","url":null,"abstract":"We announce results on a compactification of general character varieties that has good topological properties and give various interpretations of its ideal points. We relate this to the Weyl chamber length compactification and apply our results to the theory of maximal and Hitchin representations.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84641889","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 : 2020-04-30DOI: 10.1142/s0218216520500601
Ayaka Shimizu
The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.
{"title":"Prime alternating knots of minimal warping degree two","authors":"Ayaka Shimizu","doi":"10.1142/s0218216520500601","DOIUrl":"https://doi.org/10.1142/s0218216520500601","url":null,"abstract":"The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone knot diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"69 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78079260","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 : 2020-04-15DOI: 10.13137/2464-8728/30760
M. R. Casali, P. Cristofori, C. Gagliardi
The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {it semi-simple} and {it weak semi-simple crystallizations}, with a particular attention to their properties of minimizing combinatorially defined PL-invariants, such as the {it regular genus}, the {it Gurau degree}, the {it gem-complexity} and the {it (gem-induced) trisection genus}. The main theorem, yielding a summarizing result on the topic, is an original contribution. Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$-manifolds with empty or connected boundary which admit weak semi-simple crystallizations.
{"title":"Crystallizations of compact 4-manifolds minimizing combinatorially defined PL-invariants","authors":"M. R. Casali, P. Cristofori, C. Gagliardi","doi":"10.13137/2464-8728/30760","DOIUrl":"https://doi.org/10.13137/2464-8728/30760","url":null,"abstract":"The present paper is devoted to present a unifying survey about some special classes of crystallizations of compact PL $4$-manifolds with empty or connected boundary, called {it semi-simple} and {it weak semi-simple crystallizations}, with a particular attention to their properties of minimizing combinatorially defined PL-invariants, such as the {it regular genus}, the {it Gurau degree}, the {it gem-complexity} and the {it (gem-induced) trisection genus}. The main theorem, yielding a summarizing result on the topic, is an original contribution. Moreover, in the present paper the additivity of regular genus with respect to connected sum is proved to hold for all compact $4$-manifolds with empty or connected boundary which admit weak semi-simple crystallizations.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77629110","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 : 2020-04-09DOI: 10.1142/S0218216521500322
T. Fiedler
Let $M^{reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M^{reg}_n$ be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$ . We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M^{reg}_n$ by a very simple formula. This 1-cocycle depends on a natural number $a in mathbb{Z}cong H_1(V^3;mathbb{Z})$ with $0
设$M^{reg}$为不超过正则异构的长结的拓扑模空间，对于任意自然数$n > 1$设$M^{reg}_n$为在实体环面$V^3$上被一根弦环扭成一个结的框架长结的所有n根缆的模空间。我们用一个非常简单的公式将一个结的Vassiliev不变量$v_2$升级为$M^{reg}_n$的整数组合1-环。这个1-环依赖于一个自然数$a in mathbb{Z}cong H_1(V^3;mathbb{Z})$，以$0
{"title":"More 1-cocycles for classical knots","authors":"T. Fiedler","doi":"10.1142/S0218216521500322","DOIUrl":"https://doi.org/10.1142/S0218216521500322","url":null,"abstract":"Let $M^{reg}$ be the topological moduli space of long knots up to regular isotopy, and for any natural number $n > 1$ let $M^{reg}_n$ be the moduli space of all n-cables of framed long knots which are twisted by a string link to a knot in the solid torus $V^3$ . We upgrade the Vassiliev invariant $v_2$ of a knot to an integer valued combinatorial 1-cocycle for $M^{reg}_n$ by a very simple formula. This 1-cocycle depends on a natural number $a in mathbb{Z}cong H_1(V^3;mathbb{Z})$ with $0<a<n$ as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on $H_0(M^{reg}_n) times H_0(M^{reg})$ already for $n=2$.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"159 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77605223","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 : 2020-04-09DOI: 10.1142/s0218216520500911
Sandy Ganzell, A. Henrich
Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduno to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as $4n$-gons and gluing edges of these polygons, we obtain knots on surfaces that can be viewed as virtual knots. These virtual mosaics are our present objects of study. In this paper, we provide a set of moves that can be performed on virtual mosaics that preserve knot and link type, we show that any virtual knot or link can be represented as a virtual mosaic, and we provide several computational results related to virtual mosaic numbers for small classical and virtual knots.
{"title":"Virtual mosaic knot theory","authors":"Sandy Ganzell, A. Henrich","doi":"10.1142/s0218216520500911","DOIUrl":"https://doi.org/10.1142/s0218216520500911","url":null,"abstract":"Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduno to virtual knots, by including an additional tile type to represent virtual crossings. There is another interpretation of virtual knots, however, as knot diagrams on surfaces, which inspires this work. By viewing classical mosaic diagrams as $4n$-gons and gluing edges of these polygons, we obtain knots on surfaces that can be viewed as virtual knots. These virtual mosaics are our present objects of study. In this paper, we provide a set of moves that can be performed on virtual mosaics that preserve knot and link type, we show that any virtual knot or link can be represented as a virtual mosaic, and we provide several computational results related to virtual mosaic numbers for small classical and virtual knots.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"696 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74746848","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 : 2020-04-07DOI: 10.22405/2226-8383-2020-21-2-65-83
Андрей Юрьевич Веснин, Андрей Александрович Егоров
In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum volumes are realized on antiprisms and twisted antiprisms. The first 248 values of volumes of ideal right-angled polyhedra are presented. Moreover, the class of polyhedra with isolated triangles is introduces and there are obtained combinatorial bounds on their existence as well as minimal examples of such polyhedra are given.
{"title":"Идеальные прямоугольные многогранники в пространстве Лобачевского","authors":"Андрей Юрьевич Веснин, Андрей Александрович Егоров","doi":"10.22405/2226-8383-2020-21-2-65-83","DOIUrl":"https://doi.org/10.22405/2226-8383-2020-21-2-65-83","url":null,"abstract":"In this paper we consider a class of right-angled polyhedra in three-dimensional Lobachevsky space, all vertices of which lie on the absolute. New upper bounds on volumes in terms the number of faces of the polyhedron are obtained. Volumes of polyhedra with at most 23 faces are computed. It is shown that the minimum volumes are realized on antiprisms and twisted antiprisms. The first 248 values of volumes of ideal right-angled polyhedra are presented. Moreover, the class of polyhedra with isolated triangles is introduces and there are obtained combinatorial bounds on their existence as well as minimal examples of such polyhedra are given.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"146 6 1","pages":"65-83"},"PeriodicalIF":0.0,"publicationDate":"2020-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83093617","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 complex surfaces, viewed as smooth $4$-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the $2$-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to $2g+4$ for even $ggeq4$. For odd $ggeq7$, we show that the number is greater than or equal to $2g+6$. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the $2$-sphere as well.
{"title":"The number of singular fibers in hyperelliptic Lefschetz fibrations","authors":"Tulin Altunoz","doi":"10.2969/JMSJ/82988298","DOIUrl":"https://doi.org/10.2969/JMSJ/82988298","url":null,"abstract":"We consider complex surfaces, viewed as smooth $4$-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the $2$-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to $2g+4$ for even $ggeq4$. For odd $ggeq7$, we show that the number is greater than or equal to $2g+6$. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the $2$-sphere as well.","PeriodicalId":8454,"journal":{"name":"arXiv: Geometric Topology","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88343470","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：