Details

N. A'Campo developed a family of knots in the 3-sphere that satisfy many nice properties in the following way:

First view the 3-sphere as a subset of the tangent space over the plane by taking the set of tangent vectors at a point such that . This restricts to lie in the unit disk. Given a proper smooth immersion of the closed interval into the unit disk, the tangencies at each point on the curve give two points in the 3-sphere (except at the origin and at the boundary).

A'Campo describes this in http://arxiv.org/abs/math/9803081.

See also M. Hirasawa, "Visualization of A'Campo's Fibered Links and Unknotting Operation," Topology and Its Applications, 121(1–2), 2002 pp. 287–304.

One may care to generalize this Demonstration to use more than three points, use smooth curves, interact with the plot of the knot more directly, or just make it run more smoothly.