S² × [0,1] · inner sphere t = 0 · outer sphere t = 1
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Belt trick in the sphere braid group

Δ⁴ = 1 in B_n(S²)

Spinning S² by 2π drags any configuration of n points around a braid: the full twist Δ². Spinning by 4π gives Δ⁴, and that one comes undone — because the loop lifts to a closed loop in the simply connected S³.

Strands6

Rotation axis

Turns

lift ends at1

class in B_n(S²)trivial

Δ² itselforder 2

Develop the braid

Level t rises from the inner sphere to the outer one, carrying the configuration with it.

Untangle0.00

The lift in S³

Rotations at each level lift to unit quaternions starting at 1. The 4π loop closes up into a great circle, and the untangling is that circle shrinking to the basepoint. The 2π path only reaches −1, so there is no loop to contract.

shell level sphere axis slow spin