KNOT AS A COMPLETE INVARIANT OF A MORSE-SMALE 3-DIFFEOMORPHISM WITH FOUR FIXED POINTS

The paper proves that for the class G of orientation-preserving Morse–Smale 3‑diffeomorphisms with four fixed points and a unique noncompact heteroclinic curve, the equivalence class of the Hopf knot in S^2×S^1 obtained by projecting a 1‑saddle separatrix is a complete conjugacy invariant and that every Hopf class is realizable on S^3. The candidate solution defines exactly this invariant, shows its well-definedness and conjugacy invariance, proves sufficiency by lifting homeomorphisms that commute with the contraction on the sink basin and gluing across linearizing neighborhoods, and sketches realization by modifying dynamics inside the sink basin—closely mirroring the paper’s construction. Minor gaps in the candidate’s informal arguments (e.g., the equivalence of the two branches and the gluing uniqueness) are addressed more systematically in the paper via compatible foliated neighborhoods and an extension theorem. Overall, the approaches are the same in spirit and essentially the same in structure.