Nth Hull problem

From JasonCantarellaWebpage

Contents 1 Definition 2 Results 3 Visualizations 4 Online Archive of Hull Models

Definition

The nth hull of space curve C is the set of points which have the property that every plane through a point in the nth hull cuts the curve C at least 2n times.

Results

We proved in the paper The Second Hull of a Knotted Curve that every knotted curve has a nonempty 2nd hull. This leads to a new proof of the Fary-Milnor theorem, which states that every knotted curve has total curvature 4π.

Visualizations

My student John Foreman wrote code to compute n-th hulls for space polygons as polyhedra. We generated some nice pictures of second and higher hulls.

The second hull of a trefoil knot from the front.	The trefoil second hull from the side.	The 818 knot's second hull from the front.	The 818 knot's second hull from the side.
The 3rd hull of the 85 knot.	A closeup of the 3rd hull of the 85 knot.	The 818 knot's third hull from the front.	The 818 knot's third hull from the side.

Online Archive of Hull Models

A collection of our computed second hulls for some common knots is available here. The third hulls are available here. And the curves used to generate these hulls are available here.