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Brunnian links consist of $n$ linked un-knot components such that the cutting of any component leaves all components unconnected. The most famous example is the three-component Borromean rings (or links). In 1954 Milnor classified all Brunnian links up to link homotopy.

In $\mathbb{R}^3$, it is simple indeed to see that one can construct Borromean rings from identical rigid components. (The simplest case involves three identical rigid ovals---same intrinsic shape, same intrinsic size.) I have sketched six classes of rigid (un-knot) components that can be linked to form Borromean rings, and I'm sure there are an infinite number of such.

Question

Is it possible to create identical rigid components interlocked to form Brunnian links for $n \geq 4$? Is there an upper limit for $n$ for which this can be done?

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    $\begingroup$ To clarify, by rigid Brunnian link, do you require that after removing any component, the remaining unlink may be split into unknots by rigid motion of each component (say so that each component can be moved arbitrary far from the others)? This would be the natural interpretation of your terminology. $\endgroup$ – Ian Agol Apr 3 at 4:21
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    $\begingroup$ I hadn't really thought about that, but let's assume "yes." $\endgroup$ – David G. Stork Apr 3 at 4:52
  • $\begingroup$ It looks to me like you have answered your own question. If there is anything that remains, could you perhaps edit your question to highlight your remaining concerns? $\endgroup$ – Ryan Budney Apr 3 at 6:41
  • $\begingroup$ @RyanBudney: I guess I have no other concerns so I'll accept my own answer. I do wonder if this result is publishable, as an example of what one might call "Rigid knot theory" or "Homology-free knot theory." It explicitly breaks one of the fundamental tenets of classical knot theory, but it seems to me to be a sub-discipline worthy of explicit naming. It would include knot theory for Medieval chain mail and some puzzles, for instance. I haven't seen this sub-discipline explicitly identified... have you? $\endgroup$ – David G. Stork Apr 3 at 17:52
  • $\begingroup$ These kinds of examples, and other examples like it come up in several peoples' work, but they came up in slightly different contexts. My paper on splicing mentions some of these properties -- that you can find links where all the components have the same shape comes up naturally in some instances of hyperbolic links -- basically you get the result you are looking for, for free, if the link is hyperbolic with a large symmetry group. You can also describe the space of symmetry positions. Kanenobu has a paper on Brunnian properties of links that also touches on this topic. $\endgroup$ – Ryan Budney Apr 3 at 23:52
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I think this link projection shows that one can create such rigid, identical components for arbitrary $n$.

Rigid n>3 Brunnian knot

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  • $\begingroup$ It’s not clear to me that after removing one component, these can be rigidly unlinked. Do you have an idea of why this should be possible? $\endgroup$ – Ian Agol Apr 9 at 13:48
  • $\begingroup$ @IanAgol: I think that if the radius of the "loop" terminations are made extremely large, then the links can be rigidly removed (after one has been removed). I'll keep working on it. I've been looking for references on what could be called "rigid knot theory" or "homology-free knot theory," without success. Any pointers? $\endgroup$ – David G. Stork Apr 9 at 15:54

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