{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/a-computational-study-of-the-number-of","title":"A computational study of the number of connected components of positive Thompson links","arxiv_id":"2212.12556","date":"2022-12-23","proceeding":null,"authors":["Valeriano Aiello","Stefano Iovieno"],"abstract":"Almost a decade ago Vaughan Jones introduced a method to produce knots from elements of the Thompson groups $F$, which was later extended to the Brown-Thompson group $F_3$. In this article we define a way to produce permutations out of elements of the $F$ and $F_3$ that we call Thompson permutations. The number of orbits of each Thompson permutation coincides with the number of connected components of the link. We explore the positive elements of $F_3$ of fixed \\emph{width} and \\emph{height} and make some conjectures based on numerical experiments. In order to define the Thompson permutations we need to assign an orientation to each link produced from elements of $F$ and $F_3$. We prove that all oriented links can be produced in this way.","url_abs":"https://arxiv.org/abs/2212.12556v1","url_pdf":"https://arxiv.org/pdf/2212.12556v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"a-computational-study-of-the-number-of","repo_url":"https://github.com/valerianoaiello/positive-thompson-knots","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}