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This paper proves that for any link, there exists a diagram whose complementary regions can be arranged so that all regions are bounded by triangles and quadrilaterals. The argument extends earlier work on universal sequences for knots and links, relying on Eberhard-type realizability conditions and an inductive construction of suitable knot projections.",{"@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/any-link-has-a-diagram-with-only-triangles-and-quadrilaterals-main-result/657516/",{"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/any-link-has-a-diagram-with-only-triangles-and-quadrilaterals-main-result/657516.png","ImageObject",300,407,{"name":92,"@type":93},"Miles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-30","2026-10-26",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the paper prove about link diagrams?","Question",{"text":112,"@type":113},"It shows that any link has a diagram whose complementary regions can be chosen to use only triangles and quadrilaterals.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are complementary regions counted in this work?",{"text":117,"@type":113},"For a reduced connected diagram, pn counts n-gons among complementary regions, with p1 = 0 and a specific relation involving p2, p3, and higher pn.",{"name":119,"@type":110,"acceptedAnswer":120},"What role does the condition p2 = 0 play?",{"text":121,"@type":113},"The main theorem assumes p2 = 0 to guarantee that the required region-count sequence can be realized, leading to the corollary about eight triangles and quadrilaterals.","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},657516,1793123135,{"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":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"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":143,"read_time":46},13056703019404,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","arXiv :2308 . 14118v1 [math .GT] 27 Aug 2023  \nANY LINK HAS A DIAGRAM WITH ONLY TRIANGLES AND  \nQUADRILATERALS  \nREIKO SHINJO AND KOKORO TANAKA  \nDedicated to Professor Kouki Taniyama on the occasion of his 60th birthday  \nAbstract. A link diagram can be considered as a 4-valent graph embedded in the 2-sphere and divides the sphere into complementary regions. In this paper, we show that any link has a diagram with only triangles and quadrilaterals.  \nThis extends previous results shown by the authors and C. Adams.  \n1. Introduction  \nA link diagram can be considered as a 4-valent graph embedded in the 2-sphere and divides the sphere into complementary regions. Given a reduced connected diagram D, let pn (or more precisely pn (D)) be the number of n-gons of all the complementary regions of D for each n ≥ 1. Note that p 1 = 0 since D is reduced. It follows from Euler’s formula and some elementary observations that we have the following equation  \n(1) 2p2 + p3 = 8 + p5 + 2p6 + 3p7 + ··· , in which p4 does not appear; see for example [2] .  \nIn graph theory, the converse direction has been investigated, dating back to [3] . Gr¨unbaum [4] proved that any sequence {pn }n≥2,n4 of nonnegative integers with p2 = 0 that satisfies Equation (1) can be realized as a planar 4-valent 3-connected graph such that the number of its n-gon regions is pn for all n  4. This theorem is known as Eberhard’s theorem. We note that the condition p2 = 0 is atypical assumption in graph theory, since the 1-skeletons of convex polytopes are of interest. We also note that a convex 3-polytopal 4-valent graph is nothing but a connected reduced link projection without bigons. Then Jeong [5] extended this result to show that the resulting graph can be taken as a knot projection rather than a link projection.  \nIn this paper, we investigate such a problem for link diagrams; which sequence {pn }n≥2,n4 of nonnegative integers that satisfies Equation (1) can be realized asa diagram of every link such that the number of its n-gon regions is pn for all n  4? This is a continuation of the study of complementary regions of knot and link diagrams by the authors and Adams in [2] . In that paper, we introduced and investigated the notion of universal sequences for knots and links, which will be reviewed and discussed in Section 3 of this paper. See the original paper [2] or  \nDate: August 29, 2023 .  \n2020 Mathematics Subject Classification. Primary 57K10, 57M15 .  \nKey words and phrases. Knot diagram; complementary region; 4-valent graph.  \n2 REIKO SHINJO AND KOKORO TANAKA  \nthe survey [1, Chapter 10] for more details about universal sequences. The main purpose of this paper is to show the following theorem and its corollary.  \nTheorem 1.1 . For any sequence {pn }n≥2,n4 of nonnegative integers with p2 = 0 that satisfies Equation (1) and any link L, there exists a diagram DL of L such that pn (DL ) = pn for all n  4.  \nSince the sequence p2 = 0, p3 = 8 and pn = 0 (n ≥ 5) satisfies the assumption of Theorem 1.1, we have the following, which is the title of this paper:  \nCorollary 1.2 . Any link has a diagram with only eight triangles and quadrilaterals. Theorem 1.1 follows from Theorem 1.3, whose proof will be given in Section 2 .  \nTheorem 1.3 . Let P be a knot projection with the part as shown in Figure 1, where the numbers from 0 to 2 indicate the order in which the arcs are traced. Then any link L has a diagram DL such that pn (DL ) = pn (P) for all n  4.  \nFigure 1 . A part of a knot projection consisting of three strands  \nProof of Theorem 1.1 . Let {pn }n≥2,n4 be a sequence of nonnegative integers with p2 = 0 that satisfies Equation (1) . It was shown in [5] that there exists a choice of p4 and a knot projection P such that pn (P) = pn for all n ≥ 2. The proof was inductive and constructive. We recall here a rough outline of the proof in [5] . Start with the knot projection P0 in Figure 2 . It is made up of eight triangles and three quadrilaterals, and has the part as","cbCairuydmnoUUTy","https://ap.wps.com/l/cbCairuydmnoUUTy","pdf",1792786,"English","# Introduction\n## Main theorem and corollaries\n# Proof outline and construction\n## Theorems and propositions\n# Supporting lemmas","[{\"question\":\"What does the paper prove about link diagrams?\",\"answer\":\"It shows that any link has a diagram whose complementary regions can be chosen to use only triangles and quadrilaterals.\"},{\"question\":\"How are complementary regions counted in this work?\",\"answer\":\"For a reduced connected diagram, pn counts n-gons among complementary regions, with p1 = 0 and a specific relation involving p2, p3, and higher pn.\"},{\"question\":\"What role does the condition p2 = 0 play?\",\"answer\":\"The main theorem assumes p2 = 0 to guarantee that the required region-count sequence can be realized, leading to the corollary about eight triangles and quadrilaterals.\"}]","Any Link Has a Diagram with Only Triangles and Quadrilaterals - Main result | PDF",1792981742]