Optimal Rolling Trefoil

The trefoil knot is the simplest possible nontrivial knot. It can embed as a closed curve in 3-space many different ways, but some of those embeddings have the property that they are tritangentless, that is, that it has no tangent planes that intersect the curve in three places. One consequence of this is that no matter how a tritangentless trefoil sits on a table, it will never touch the table in more than two places. It also means that the knot rocks back and forth easily, and rolls down even very shallow inclines.

Equations for tritangentless trefoils can be found in Morton’s paperĀ Trefoil Knots Without Tritangent Planes. By changing parameters in these equations we get different tritangentless conformations. Some of these conformations roll better than others; which one is optimal? And can we prove that one is optimal?

Students working on this project will collaborate with Dr. Lucas and Dr. Taalman, and draw on expertise from other faculty members. We will experiment with 3D printed conformations of tritangentless trefoils, and also try to use calculus, linear algebra, and geometry to arrive at an optimal solution. If one is found, a short research paper would likely be the result.