What happens:
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An approximation of an ideal trefoil is embeded in R^4 and very slightly perturbed to break the symmetries.
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Using a 4D version of the ropelength minimizing algorithm SONO
(by P. Pieranski, In Search of Ideal Knots in [SKK98], 20-41; implemented by M. Carlen. and adopted to R^4 by H. Gerlach) the knot gets untangled in R^4.
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The movie shows the projection back to the embeded R^3. The final circle does not lie in this subspace, so it becomes an elipse.
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This numerical experiment was proposed by Maciej Borodzik.
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The second video encodes the 4th dimension as color or temperature (thanks to Philipp Reiter to remind me of this
visualization trick).
Download the movies (License
cc-by-sa-3.0):
r4_untangle.avi
r4_untangle_color.avi
Rendered with
POV-Ray.
Valid
xhmtl.