Do quantum linked rings of quantum knots have energy? There has been an experiment where infinite quantum linked rings were created to make a quantum knot.The structure is topologically stable and the knot can't be untied without breaking the rings. 
If these rings have energy and allow the structure to be stable, wouldn't there be infinite energy and thus wouldn't we need infinite energy to break these rings?

For more details, see


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*Scientists tie quantum materials into infinite knots, Steve Dent, Engadget, 01.20.16 


The original paper is at


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*Tying quantum knots. DS Hall et al. Nature Phys 12, 478–483 (2016), arXiv:1512.08981.

 A: This is a bit of hyperbole on the part of the authors of the original paper, which wasn't very well processed by the author of the engadget piece you read. Your original inference ('if there are infinite rings and each ring has finite energy, there is infinite total energy') is reasonable enough, but that's not what's happening.
The processes involved are best examined in this figure from the original paper:

Here you can see the knot as it looks like in real space, with a bunch of rings, on the top left, and at bottom right you have a parametrization of the knot in some funky mathematical space called $S^2$ whose precise nature is beside the point. 
Each ring on real space corresponds to one of the segments on $S^2$, through the colour coding, and as you can see the figure only shows a subset of the possible rings. In its full glory, the experiment actually has rings corresponding to all the points on the equator of the sphere in $S^2$, and in real space when you display all the rings that are present they fill the gaps to make a full donut-shaped surface surrounding the white ring.
Now, the thing with energy is that there is a finite amount of energy, and it is shared equally by all the points on that equator. This means that if you select a stretch of that equator spanning an angle $\Delta \theta$ (like e.g. the greenish segment at the end of the white arrow), then it will contain an energy $\Delta E = \kappa \,\Delta\theta$ proportional to the length $\Delta \theta$ of the arc, where $\kappa$ is a constant. Thus, there are infinite  rings, corresponding to infinite points along the equator, but each individual ring (much thinner than the ones displayed) has an infinitesimal amount of energy.
