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2025-10-29 01:21:19 -

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (2)
2. Theoretical Foundation: Topological Invariants 2.1 Linking Number (Lk) For two simple, unknotted but mutually entwined closed curves (like an ideal Hopf link), the linking number Lk is a topological invariant representing the total number of times (with sign) one loop winds around the other. For two vortex rings, Lk quantitatively describes the topological strength of their mutual...

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2025-10-29 01:30:19 -

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (3)
4. Superposition of Counter-rotating Vortex Rings and Knot Formation The situation is radically different when two vortex rings with opposite circulation directions (counter-rotating) interact: (1) Topological Description: Can be seen as the superposition of a positive Hopf link and a negative Hopf link. The total topological charge (net circulation) of the system is zero. (2) Dynamical...

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2025-10-29 01:37:08 -

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (4)
5. Discussion and Outlook The framework constructed in this study abstracts specific physical phenomena into topological operations: a. Co-rotating superposition ≈ Addition of topological charges, structure tends towards stability or higher-order linking. b. Counter-rotating superposition ≈ Subtraction of topological charges, structure tends towards generating singular knots or...

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2025-10-29 01:45:15 -

A Study on Topological Vortex Ring Interactions Based on Möbius Loop and Hopf Link Concepts (5)
References Călugăreanu, G. (1959). L'intégrale de Gauss et l'analyse des nœuds tridimensionnels [The Gauss Integral and the Analysis of Three-Dimensional Knots]. Revue de mathématiques pures et appliquées, 4, 5-20. Faddeev, L. D., & Niemi, A. J. (1997). Stable knot-like structures in classical field theory. Nature, 387(6628), 58-61. Hopf, H. (1931). Über...

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