{"title":"Application and implication of knot theory to the circular restricted three-body problem","authors":"Mason R. Mill, Robert A. Bettinger","doi":"10.1007/s10509-025-04469-w","DOIUrl":null,"url":null,"abstract":"<div><p>This paper investigates the application of knot theory to the classification of orbit families in the Circular Restricted Three-Body Problem (CR3BP). Motivated by the infinite variety of possible orbits—many of which remain unnamed and uncataloged—this paper applies polynomial knot invariants, primarily the Alexander polynomial, to establish a relation between knot structures and orbital trajectories. An algorithm is developed to extract knot types from three-dimensional trajectories enabling the identification and differentiation of complex orbit families. Knot theory topics explored and correlated to CR3BP trajectories include the torus knot and unknot. The findings provide a novel topological framework for understanding CR3BP dynamics, offering both theoretical understanding and practical modeling in astrodynamics for multi-body gravitational systems.</p></div>","PeriodicalId":8644,"journal":{"name":"Astrophysics and Space Science","volume":"370 8","pages":""},"PeriodicalIF":1.8000,"publicationDate":"2025-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10509-025-04469-w.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Astrophysics and Space Science","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1007/s10509-025-04469-w","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"ASTRONOMY & ASTROPHYSICS","Score":null,"Total":0}
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Abstract
This paper investigates the application of knot theory to the classification of orbit families in the Circular Restricted Three-Body Problem (CR3BP). Motivated by the infinite variety of possible orbits—many of which remain unnamed and uncataloged—this paper applies polynomial knot invariants, primarily the Alexander polynomial, to establish a relation between knot structures and orbital trajectories. An algorithm is developed to extract knot types from three-dimensional trajectories enabling the identification and differentiation of complex orbit families. Knot theory topics explored and correlated to CR3BP trajectories include the torus knot and unknot. The findings provide a novel topological framework for understanding CR3BP dynamics, offering both theoretical understanding and practical modeling in astrodynamics for multi-body gravitational systems.
期刊介绍：
Astrophysics and Space Science publishes original contributions and invited reviews covering the entire range of astronomy, astrophysics, astrophysical cosmology, planetary and space science and the astrophysical aspects of astrobiology. This includes both observational and theoretical research, the techniques of astronomical instrumentation and data analysis and astronomical space instrumentation. We particularly welcome papers in the general fields of high-energy astrophysics, astrophysical and astrochemical studies of the interstellar medium including star formation, planetary astrophysics, the formation and evolution of galaxies and the evolution of large scale structure in the Universe. Papers in mathematical physics or in general relativity which do not establish clear astrophysical applications will no longer be considered.
The journal also publishes topically selected special issues in research fields of particular scientific interest. These consist of both invited reviews and original research papers. Conference proceedings will not be considered. All papers published in the journal are subject to thorough and strict peer-reviewing.
Astrophysics and Space Science features short publication times after acceptance and colour printing free of charge.
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