Theoretical analysis method of filament bending–twisting energy and entanglement degree in forming α filament knotoid

Knots are commonly used for binding, securing objects, and decorative purposes in daily life. Many geometric structures of applicable filament knots can be obtained from knot theory. The homeomorphic structure of an α knotoid can be derived from the topological invariants of equivalent knots and α filament knotoid. The geometric topology of an α knotoid is analyzed by Alexander’s theorem, and the entanglement degree of an α filament knotoid is determined. A corresponding relationship is obtained between the bending and twisting energy of the knot and the geometric parameters of the simplified space curve. The theoretical foundation for analyzing the motion of corresponding knotting mechanisms is based on research findings regarding the topological geometry of α knotoid. Research results on knot entanglement degree provide a method for determining the complexity of the filament knot. Theoretical derivation methods and conclusions about filament bending and twisting energy are vital for confirming the power requirements in designing automatic forming equipment for filament knots. The results show that the entanglement degree of α knotoid filament is S ₀ = 6. The effectiveness of the bending and twisting energy model in practical applications was verified by comparing special treated hair filaments with ordinary hair filaments. The maximum instantaneous bending–twisting energy E ₀ in the design process increases with the diameter D and decreases as the elasticity modulus of the material decreases. A reasonable knotting process for α filament knotoid was designed based on its analysis of entanglement degree and minimum energy during knotting.