To see a curve on the three-sphere rotating in red and white, activate the movie
To see a curve on the three-sphere rotating with nice colors but low resolution, activate the movie
A much nicer movie shows the curve and its pushoffs as a series of colored bands.
Another movie shows the curve "pass through infinity" in three-space by passing through the center of projection in four-space.
Self-Linking for Curves in Three-Space
Letter to Kuiper on Torus Knots on the Three-Sphere