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

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.