It is well known that there are trefoil knots without tritangent planes, and with 3d printers one can print these beautiful objects and make them roll on planes. (An example:https://www.youtube.com/watch?v=IpSsOfe5dMk (8:50) )

At any instant of time there are exactly 2 points of contact between the knot and the plane. I'm curious about (assuming rolling without slipping) how one can compute the trajectory of contact points, the trajectory of the center of mass, etc.

Does anyone know how to compute such things or give a reference about this sort of problem?

(I have printed a model based on the torus knots described by Morton (1991) and the trajectories of the contact points look like "deformed cycloids")

  • $\begingroup$ possibly related: mathoverflow.net/questions/180194/… $\endgroup$ – Ben Crowell May 19 '15 at 23:27
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    $\begingroup$ The relevant material seems to be at the end of the talk, around 9:00? One thing to think about would be whether you want to ask these questions about knots with ideal shapes, or knots that are arbitrary fattened embeddings of the topological knot in 3-space. Ideal shapes are poorly characterized: arxiv.org/abs/1402.5760 . It's not obvious to me whether the dynamics are Hamiltonian. See, e.g., en.wikipedia.org/wiki/Chaplygin_sleigh . $\endgroup$ – Ben Crowell May 19 '15 at 23:35

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