In musing about untangles, I thought it might be best to start with the simplest one that can’t be flattened. I am not claiming that more complicated tangles can be drawn on the surface of a sphere! Far from it. Jim On Thursday, July 19, 2018, Dan Asimov <dasimov@earthlink.net> wrote:

I'm not sure I understand the part about mandating the optima in advance.

If the tangles are well-tangled in space, I'm not sure how an optimum could be able to be a circle arc.

(Further, aren't tangles supposed to happen *inside* something, like a cube?)

—Dan

Jim Propp wrote: ----- Consider the points A=(1,0,0), B=(-1,0,0), C=(0,1,0), and D=(0,-1,0). There are lots of ways to draw two disjoint 3D arcs joining A with B and C with D. What if we want them to be short without getting too close to each other? We might set up some “short-but-not-too-close” objective functional and optimize it using calculus of variations. There are probably many ways to do this, some solvable and some not. Does anyone know of work along these lines?

In the spirit of calculus-of-variations-of-variations, we might mandate the optima in advance (each arc is a half-circle on the sphere x^2+y^2+z^2=1, one in the upper hemisphere and one in the lower hemisphere) and then ask, For what objective functional is this pair of arcs locally optimal? ... -----

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