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Encountered this knot in a real life example (tangled necklace). Is it possible to untangle it (without e.g cutting the loop) and if so, how?

Knot

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    $\begingroup$ What is the question? $\endgroup$
    – justhalf
    Dec 22, 2022 at 5:39
  • $\begingroup$ @justhalf I post a picture of a knot. What do you think the question is? $\endgroup$
    – Tdonut
    Dec 22, 2022 at 8:28
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    $\begingroup$ There could be multiple possibilities. You could've been asking about step-by-step how to untangle it, or whether it is possible to shape an uncut loop of string in the shape you showed, or whether it is isomorphic to some other shape with N intersections, or whether you have transcribed this correctly and would like to get feedback where the error would have been. Or, you know, maybe there is no question at all since you just find this shape cool. :) $\endgroup$
    – justhalf
    Dec 22, 2022 at 10:24
  • $\begingroup$ The question has been edited to ask explicitly whether the loop can be untangled without cutting. It seems to me that this resolves the objection that led to the question's closing, but three users left it closed when it was presented in review. I am reluctant to overrule the Will Of The People by reopening it myself, but I don't quite understand why reopening wasn't successful. (It's possible that it might then get closed again for being not puzzle-y enough, but that's a separate issue.) $\endgroup$
    – Gareth McCaughan
    Dec 24, 2022 at 12:19
  • $\begingroup$ It got 4 reopen votes from other people, so I added mine. Again, I can't guarantee that it won't now get re-closed for being more a mathematical exercise than a puzzle or something of the kind. $\endgroup$
    – Gareth McCaughan
    Dec 27, 2022 at 18:14

1 Answer 1

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First, to check that it's possible for this knot diagram to be untangled (that this is a diagram of the unknot), I computed its Jones Polynomial. Its Jones Polynomial evaluated to 1, which is the value of all unknot diagrams. For all known knot diagrams which cannot be untangled (which are not diagrams of the unknot), the Jones polynomial is different from 1. Therefore, at this point I was confident that this knot could be untangled.

Here's how I untangled it. Read left-to-right, then top-to-bottom:

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  • $\begingroup$ The way you did your first step requires a longer string than is given. Since the question noted that this is a physical object, I don't think this is a good solution. $\endgroup$
    – msh210
    Jan 11, 2023 at 11:36
  • $\begingroup$ At least according to Wikipedia, it's still unproven that only the unknot has trivial Jones polynomial. $\endgroup$
    – msh210
    Jan 11, 2023 at 11:38
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    $\begingroup$ @msh210 I think the diagrams show the arrangement of the string over time. It’s not necessarily to scale. Are you suggesting that something about the loop of string makes it impossible to exist in that configuration (i.e. with that pattern of crossings)? $\endgroup$
    – Sneftel
    Jan 11, 2023 at 15:03
  • $\begingroup$ No, @Sneftel, just that it was drawn longer. Anyway, on second thought, I've upvoted this now, though I still maintain it doesn't meet the letter of the question. $\endgroup$
    – msh210
    Jan 11, 2023 at 16:08
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    $\begingroup$ @msh210 On the Jones Polynomial, that's why I said "all known knot diagrams ..." - though if this was the first counterexample to the pattern, that would be way more exciting. $\endgroup$
    – isaacg
    Jan 11, 2023 at 17:56

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