Application and implication of knot theory to the circular restricted three-body problem

• Published in: Astrophysics and space science Vol. 370; no. 8; p. 77
• Main Authors: Mill, Mason R., Bettinger, Robert A.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.08.2025; Springer Nature B.V
• Subjects: Astrobiology; Astrodynamics; Astronomy; Astrophysics and Astroparticles; Cosmology; Euclidean space; Knot theory; Knots; Observations and Techniques; Orbits; Physics; Physics and Astronomy; Polynomials; Space Exploration and Astronautics; Space Sciences (including Extraterrestrial Physics; Three body problem; Toruses; Trajectories; Alexander polynomial; Knot theory; Torus knot; Unknot
• ISSN: 0004-640X; 1572-946X
• DOI: 10.1007/s10509-025-04469-w

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.