On detection probabilities of link invariants

On detection probabilities of link invariants | 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 On detection probabilities of link invariants Tuomas Kelomäki, Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8667815/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 We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. We also use a big data approach to analyze knots and provide evidence that, for knots as well, these invariants exhibit the same asymptotic failure of detection. Full Text Additional Declarations No competing interests reported. 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. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-8667815","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":587919416,"identity":"f9594bbe-3071-42d9-8b75-22de1a761f8f","order_by":0,"name":"Tuomas Kelomäki","email":"","orcid":"","institution":"Aalto University","correspondingAuthor":false,"prefix":"","firstName":"Tuomas","middleName":"","lastName":"Kelomäki","suffix":""},{"id":587919418,"identity":"7f532932-120d-4d75-b228-fc353e842718","order_by":1,"name":"Abel Lacabanne","email":"","orcid":"","institution":"University of Clermont Auvergne","correspondingAuthor":false,"prefix":"","firstName":"Abel","middleName":"","lastName":"Lacabanne","suffix":""},{"id":587919420,"identity":"524ac801-35f5-46ea-b687-c549f68ac265","order_by":2,"name":"Daniel Tubbenhauer","email":"data:image/png;base64,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","orcid":"","institution":"The University of Sydney","correspondingAuthor":true,"prefix":"","firstName":"Daniel","middleName":"","lastName":"Tubbenhauer","suffix":""},{"id":587919422,"identity":"3f8b49cb-4f0c-4264-9bdb-2a770a142d02","order_by":3,"name":"Pedro Vaz","email":"","orcid":"","institution":"Université Catholique de Louvain","correspondingAuthor":false,"prefix":"","firstName":"Pedro","middleName":"","lastName":"Vaz","suffix":""},{"id":587919423,"identity":"c5322fbc-cadc-4910-985d-dafa94ca5fa2","order_by":4,"name":"Victor L. 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