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? What got me wondering along these lines was an object I saw in the halls of the Dartmouth math department, apparently implementing Conway’s rational tangles game. It made me wonder whether there’s a “best” way to embed any given tangle. Jim Propp