2
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Here are some my attempts.

Show[ParametricPlot3D[
  RotationMatrix[1/2 u, {1, 0, 0}].{Sin[u], Cos[u], 0}, {u, 0, 
   4 \[Pi]}, PlotRange -> 1.1], Graphics3D[{Opacity[0.5], Sphere[]}]]

enter image description here

Show[ParametricPlot3D[
  RotationMatrix[1/3 u, {1, 0, 0}].{Sin[u], Cos[u], 0}, {u, 0, 
   6 \[Pi]}, PlotRange -> 1.1], Graphics3D[{Opacity[0.5], Sphere[]}]]

enter image description here

I would like the curve to be more symmetric and so that at one point there is only one intersection with itself as opposed to image 2. By symmetric I mean that the curve should cover surface of sphere evenly as possible. I know that there are only five platonic solids that are perfectly symmetric so I do not insist on a such perfection. But, say, plotting curve that resembles the following knot would be OK for me.

knot

Notice that it is just one closed loop wrapped around sphere, it is just doubled with two differently colored ropes.

Any ideas?

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  • $\begingroup$ Relevant: youtube.com/watch?v=ssZMcKS9Evc $\endgroup$ – Henrik Schumacher Jan 14 '19 at 20:01
  • $\begingroup$ @Henrik Schumacher Thank you, I only uploaded first image found by google, have not realized it was from youtube. So maybe the video can help to parametrize it somehow although there are not specified any mathematical properties of it. But I would be happy with parametrization of any type of such knot, even simpler one. $\endgroup$ – azerbajdzan Jan 14 '19 at 21:03

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