[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-665657-105":59,"doc-detail-665657-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","rectangular-diagrams-of-foliations","RECTANGULAR DIAGRAMS OF FOLIATIONS","","A concept of rectangular diagrams for codimension-one foliations in the three-sphere is introduced. The work proves that any co-orientable finite-depth foliation in the complement of a link admits a presentation by a rectangular diagram compatible with the rectangular diagram of the link. The construction extends rectangular-diagram formalisms previously developed for links and surfaces, encoding both combinatorial data and holonomy maps within a single diagram.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/rectangular-diagrams-of-foliations/665657/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/rectangular-diagrams-of-foliations/665657.png","ImageObject",300,407,{"name":92,"@type":93},"Cart","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-29",true,{"@type":102,"interactionType":103,"userInteractionCount":14},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What is the main contribution of this paper?","Question",{"text":112,"@type":113},"It introduces a rectangular-diagram presentation for codimension-one foliations in the complement of links in S3, extending earlier diagram formalisms for links and surfaces.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What does it mean for a foliation to be represented by a rectangular diagram?",{"text":117,"@type":113},"The diagram encodes the information needed to reconstruct the foliation, including combinatorial data and holonomy maps, compatible with a given diagram of the link.",{"name":119,"@type":110,"acceptedAnswer":120},"Why are cavities and branching leaves introduced?",{"text":121,"@type":113},"Because a foliation cannot be normalized by simultaneously normalizing all its leaves, the method allows finitely many leaves to branch and create cavities, which are then resolved by deformations that “de-fate” the cavities.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},665657,1793305510,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":14,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":36,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":143},18829141979164,"https://eur-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","arXiv :2508 .07732v1 [math .GT] 11 Aug 2025  \nRECTANGULAR DIAGRAMS OF FOLIATIONS  \nMIKHAIL CHERNAVSKIKH, IVAN DYNNIKOV  \nAbstract. A concept of a rectangular diagram of a foliation in the three-sphere is introduced. It is shown that any co-orientable 􀀌nite depth foliation in the complement of a link admits a presentation by  \na rectangular diagram compatible with the given rectangular diagram of the link.  \nContents  \n1. Introduction 1  \n2. Rectangular diagrams of links and (quasi-)surfaces 3  \n3. Deformations of rectangular diagrams of quasi-surfaces 7  \n4. Thin tubes around a link 9  \n5. Packs of rectangles 11  \n6. Representing a Reeb component 13  \n7. Cavities 16  \n8. Representing a foliation by a rectangular diagram 20  \n9. Examples 23  \n10. Normalizing a foliation 24  \n11. General 􀀌nite depth foliations on S3 39  \nReferences 40  \n1. Introduction  \nThe aim of this paper is to propose a systematic way to represent codimension-one foliations in link complements in the three-sphere. Among such foliations, taut ones have received most attention in the literature.  \nRecall that a codimension-one foliation in a compact three-manifold M is taut if, for any leaf F of the foliation, there is a closed curve in M intersecting F and transverse to the foliation. If M has a non-empty boundary ∂M , one typically requires that the foliation is transverse to ∂M. The boundary ∂M then must be a union of tori.  \nThe importance of taut foliations comes, in particular, from the fact that they can be used to detect the Seifert genus of a link. Namely, a Seifert surface of a non-split link can be included as a leaf into ataut foliation in the link complement if and only if it has smallest possible genus. In the ‘only if’ direction this is a result of W. Thurston [14], and the converse was proved by D. Gabai [8] . If a link is split, then, by Novikov’s theorem [11], its complement supports no taut foliation.  \nIn order to use taut foliations in applications, it would be convenient to have a uniform way of representing them by diagrams. Since foliations are quite complicated objects, this is not obvious how todo, and to date, such a way has not been worked out. There are two mutually related techniques that are commonly used to con􀀌rm the existence of a taut foliation in the given three-manifold: sutured manifold hierarchies and branched surfaces. Such objects are usually presented by a few pictures accompanied by a verbal description that contain some combinatorial information about the foliation structure, but don’t specify a unique foliation up to isotopy.  \nThe method proposed in this paper for presenting foliations allows one to put into a single diagram all information needed to recover a foliation, which includes not only combinatorial data, but also holonomy  \n2 MIKHAIL CHERNAVSKIKH, IVAN DYNNIKOV  \nmaps. We achieve this by extending to foliations the formalism of so called rectangular diagrams developed previously for links and surfaces in the three-sphere.  \nThe idea behind the rectangular diagram formalism is that we can represent one-dimensional objects in S3 such as links and graphs by 􀀌nite families of points in the two-torus T2 . To do so, we view S3 as a join of two circles, which is, by de􀀌nition, a quotient space of T2 ×[0;1] . So, every point of T2 represents an arc in S3 . In this way, by following certain rules, we can represent the one-dimensional skeleton of a surface so that the entire surface is recovered uniquely. Once we have managed to represent individual surfaces by collections of points in T2 , it is natural to try to represent one-parametric families of surfacesand foliations by collections of curves.  \nThere are, however, two technical di􀀎culties in realizing this idea. First, in order to be presented by a rectangular diagram, a surface must be ‘normalized’. Normalizing an individual surface is not hard, but normalizing all leaves of a foliation simultaneously is never possible. To overcome this, we allow 􀀌nitely ma","cbCaiefVh8J1hhR7","https://ap.wps.com/l/cbCaiefVh8J1hhR7","pdf",733361,"English","# Contents\n## Introduction\n## Rectangular diagrams of links and (quasi-)surfaces\n## Deformations of rectangular diagrams of quasi-surfaces\n## Thin tubes around a link\n## Packs of rectangles\n## Representing a Reeb component\n## Cavities\n## Representing a foliation by a rectangular diagram\n## Examples\n## Normalizing a foliation\n## General finite depth foliations on S3\n## References","[{\"question\":\"What is the main contribution of this paper?\",\"answer\":\"It introduces a rectangular-diagram presentation for codimension-one foliations in the complement of links in S3, extending earlier diagram formalisms for links and surfaces.\"},{\"question\":\"What does it mean for a foliation to be represented by a rectangular diagram?\",\"answer\":\"The diagram encodes the information needed to reconstruct the foliation, including combinatorial data and holonomy maps, compatible with a given diagram of the link.\"},{\"question\":\"Why are cavities and branching leaves introduced?\",\"answer\":\"Because a foliation cannot be normalized by simultaneously normalizing all its leaves, the method allows finitely many leaves to branch and create cavities, which are then resolved by deformations that “de-fate” the cavities.\"}]","RECTANGULAR DIAGRAMS OF FOLIATIONS | PDF",101]