{"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/patterns-in-knot-floer-homology","title":"Patterns in Knot Floer Homology","arxiv_id":"2307.03297","date":"2023-07-06","proceeding":null,"authors":["Ekaterina S. Ivshina"],"abstract":"Based on the data of 12-17-crossing knots, we establish three new conjectures about the hyperbolic volume and knot cohomology: (1) There exists a constant $a \\in R_{>0}$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \\to \\infty$: $\\log r(K) < a \\cdot Vol(K)$ for a knot $K$ where $r(K)$ is the total rank of knot Floer homology (KFH) of $K$ and $Vol(K)$ is the hyperbolic volume of $K$. (2) There exist constants $a,b\\in R$ such that the percentage of knots for which the following inequality holds converges to 1 as the crossing number $c \\to \\infty$: $\\log \\det(K) < a \\cdot Vol(K) + b$ for a knot $K$ where $\\det(K)$ is the knot determinant of $K$. (3) Fix a small cut-off value $d$ of the total rank of KFH and let $f(x)$ be defined as the fraction of knots whose total rank of knot Floer homology is less than $d$ among the knots whose hyperbolic volume is less than $x$. Then for sufficiently large crossing numbers, the following inequality holds: $f(x)<\\frac{L}{1+\\exp{(-k \\cdot (x-x_0))}} + b$ where $L, x_0, k, b$ are constants.","url_abs":"https://arxiv.org/abs/2307.03297v2","url_pdf":"https://arxiv.org/pdf/2307.03297v2.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":"patterns-in-knot-floer-homology","repo_url":"https://github.com/eivshina/patterns-in-knot-floer-homology","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}