N. A'Campo describes a manner of producing a knot in the 3-sphere from a proper immersion of an arc in the unit disk. Here is a version that uses a polygonal path in a square to form a "long" knot in 3-space. The 2D slider at the top left changes the viewing angle. Move the three locators to change the polygonal path.
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).