Digit topology circles and their isomorphism identification for deriving topology graphs and creating complex closed mechanisms

Digit topology circles and their isomorphism identification for deriving topology graphs and creating complex closed mechanisms | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Digit topology circles and their isomorphism identification for deriving topology graphs and creating complex closed mechanisms Yi Lu, Yang Lu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4701963/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The topology graph is an effective tool for creating various complex mechanisms and machines. However, the derivation and the isomorphism identification of the topology graphs are quite complicated because topology graphs may include (binary, ternary,...) links. A digit topology circle C j ( k ) i excluding (binary, ternary) links is proposed and studied for simplifying the derivation and isomorphism identification of the topology graph. First, the conceptions of C j ( k ) i are explained and their derivations are studied. Second, a method of the character strings is proposed and studied for representing C j ( k ) i and reducing the isomorphism C j ( k ) i , and many different C j ( k ) i are represented by different digit lines and digit arcs based on the proposed method, and many different C j ( k ) i are constructed. Third, a rule for identifying isomorphism C j ( k ) i is discovered and proved using the connections in the digit topology circle, and some isomorphism C j ( k ) i are identified from constructed different C j ( k ) i . Furth, many new different C j ( k ) i with the ternary link are derived by adding ternary link into C j ( k ) i . Finally, a closed mechanism of 6-DOF the hybrid machine tool/hybrid grasper is created by combining 37 binary links with the new digit topology circle C 3(2)10 + ternary link; two closed mechanisms of the 6-DOF leg-foot are created by combining 46 binary links with digit topology circles C 26(1) i ( i = 101,104), respectively. digit topology circle topology graph type synthesis closed mechanism isomorphism identification basic link Full Text Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4701963","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":335817532,"identity":"77bd4af4-150b-4e5d-8b04-d4a64644fa87","order_by":0,"name":"Yi Lu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAq0lEQVRIiWNgGAWjYBACPiA+8AHCNiBOCxsQH5xBshZmHtK0SOQYHrZtq0tsYG/eJsFQc4coLQaHc9sOJzbwHCuTYDj2jGgtBxIbJHLMJBgbDhOpxRLkMPk3pGhhbGMG2sJDrBaeZwUHe84dNm7jSSu2SDhGhBZ+9uTNH36U1cn2sx/eeONDDRFaGBg4DBgY2SARxJBAjAYGBvYHDAx/iFM6CkbBKBgFIxQAAPg2NfNCvOZ8AAAAAElFTkSuQmCC","orcid":"","institution":"Yanshan University","correspondingAuthor":true,"prefix":"","firstName":"Yi","middleName":"","lastName":"Lu","suffix":""},{"id":335817533,"identity":"e3720c26-2fd9-40ef-ba76-29e0a428b520","order_by":1,"name":"Yang Lu","email":"","orcid":"","institution":"Yanshan daxue: Yanshan University","correspondingAuthor":false,"prefix":"","firstName":"Yang","middleName":"","lastName":"Lu","suffix":""}],"badges":[],"createdAt":"2024-07-08 01:33:35","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4701963/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4701963/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":70284680,"identity":"487fff01-d661-482b-b625-640fb93c93a3","added_by":"auto","created_at":"2024-12-01 16:05:42","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":466878,"visible":true,"origin":"","legend":"","description":"","filename":"digittopologycirclefor.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4701963/v1_covered_b8afffe0-af7f-4f02-a573-142861c81880.pdf"}],"financialInterests":"","formattedTitle":"Digit topology circles and their isomorphism identification for deriving topology graphs and creating complex closed mechanisms","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"digit topology circle, topology graph, type synthesis, closed mechanism, isomorphism identification, basic link","lastPublishedDoi":"10.21203/rs.3.rs-4701963/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4701963/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe topology graph is an effective tool for creating various complex mechanisms and machines. However, the derivation and the isomorphism identification of the topology graphs are quite complicated because topology graphs may include (binary, ternary,...) links. A digit topology circle \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e excluding (binary, ternary) links is proposed and studied for simplifying the derivation and isomorphism identification of the topology graph. First, the conceptions of \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e are explained and their derivations are studied. Second, a method of the character strings is proposed and studied for representing \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e and reducing the isomorphism \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e, and many different \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e are represented by different digit lines and digit arcs based on the proposed method, and many different \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e are constructed. Third, a rule for identifying isomorphism \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e is discovered and proved using the connections in the digit topology circle, and some isomorphism \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e are identified from constructed different \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e. Furth, many new different \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e with the ternary link are derived by adding ternary link into \u003cem\u003eC\u003c/em\u003e\u003csub\u003e\u003cem\u003ej\u003c/em\u003e(\u003cem\u003ek\u003c/em\u003e)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e. Finally, a closed mechanism of 6-DOF the hybrid machine tool/hybrid grasper is created by combining 37 binary links with the new digit topology circle \u003cem\u003eC\u003c/em\u003e\u003csub\u003e3(2)10\u003c/sub\u003e + ternary link; two closed mechanisms of the 6-DOF leg-foot are created by combining 46 binary links with digit topology circles \u003cem\u003eC\u003c/em\u003e\u003csub\u003e26(1)\u003cem\u003ei\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003ei\u003c/em\u003e\u0026thinsp;=\u0026thinsp;101,104), respectively.\u003c/p\u003e","manuscriptTitle":"Digit topology circles and their isomorphism identification for deriving topology graphs and creating complex closed mechanisms","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-08-30 06:26:10","doi":"10.21203/rs.3.rs-4701963/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"754e0f17-0521-47ad-8ff5-0aae64300723","owner":[],"postedDate":"August 30th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-12-01T16:05:30+00:00","versionOfRecord":[],"versionCreatedAt":"2024-08-30 06:26:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4701963","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4701963","identity":"rs-4701963","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}