The Hopf map η: S³ → S² sends each point on the 3-sphere to a point on the 2-sphere. Each fiber η⁻¹(p) is a circle S¹. Any two distinct fibers are linked! The fibration is a principal U(1) bundle and underlies the quantum Hall effect.
Base point θ (S²)1.00
Base point φ (S²)0.00
Fiber count12
Rotation speed0.5
Perspective3.0
Hopf map: η(z₁,z₂) = (2Re(z₁z̄₂), 2Im(z₁z̄₂), |z₁|²−|z₂|²) for (z₁,z₂) ∈ ℂ² with |z₁|²+|z₂|² = 1
Each fiber is a great circle in S³, projected to ℝ³ via stereographic projection → a torus link