In https://plus.google.com/108432079989441783124/posts/LHewqvcj5Xo T. Abderrahman explains what Borromean rings are. As I noticed in a comment, the underlying order structure is the same as in Condorcet's paradox, and as a former student in physics, I wonder if this structure could explain why in quantum chromodynamics, quarks appear in triplets where each element has its own "color" (precisely "red", "green, and "blue", so that we only observe "white" particles). Is there thus a representation theoretic (as elementary particles are irreducible representations of Lie groups) manifestation of this peculiar order structure ?

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    $\begingroup$ This does not seem like a well-defined mathematical question, and so is not in the scope of mathoverflow. I'm voting to close it. $\endgroup$ – Ian Agol Sep 3 '18 at 23:21
  • $\begingroup$ Would it then be in MSE's scope ? $\endgroup$ – Sylvain JULIEN Sep 3 '18 at 23:24
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    $\begingroup$ I don't think so, it's not a precisely formulated mathematical question. Moreover, your view that each ring lies over the next is an artifact of the projection, and is not intrinsic to the Borromean rings (a different projection will provide a different pairwise order). So I don't see any way that your question could possibly be refined to make a mathematically precise question. $\endgroup$ – Ian Agol Sep 3 '18 at 23:31
  • $\begingroup$ In general, a lot of different things happen in different in small dimensions. 3 dimensions is the smallest number of dimensions where a random walk doesn't return you to the origin with probability 1. 3 is the first odd prime and the only prime which is 1 less than a perfect square. 3 is the smallest k where k-SAT is NP-complete. 3 is the smallest dimension where the Banach-Tarski decomposition can occur. Etc. One can make similar lists for n=1,2,4,5,6, at least. Connecting to specific things in n dimensions takes a lot more than an essentially superficial resemblance. $\endgroup$ – JoshuaZ Sep 4 '18 at 2:25

Brunnean links--in which cutting any component knot leads to the separation of all component knots--exist for any $n>0$ (not just $n=3$ for Borromean links) and Condorcet's paradox also holds for any $n>2$, so any possible relation between them is irrelevant to the specific number three of colors in quarks of interest in the question.

  • $\begingroup$ Nothing wrong with the Brunnean link for n=1 either that is just the unknot. Remove that unknot and the set of remaining components is empty and hence is vacuously a set of unlinked knots. $\endgroup$ – JoshuaZ Sep 4 '18 at 1:21
  • $\begingroup$ @JosuaZ: Yep.... thanks. So edited. $\endgroup$ – David G. Stork Sep 4 '18 at 2:10

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