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It is an open question as to whether there is a polynomial time algorithm for recognizing the unknot.

Consider the following procedure for doing so on an actual physical string: Suppose there is a physical string that is tangled and I am holding both of its ends. To determine whether the string is knotted, all I have to do is pull on both ends, tightening the string. If we end up with an unknotted string, then the string is unknotted. Otherwise, the string is knotted. I would think if we simulated this process on a digital computer, it would take polynomial time, since in real life it is quick, at least in my experience. Has this idea ever been considered in the literature?

Update: When I wrote this question, I made a mistake in my understanding of what is the unknot. The mathematical definition is a string with its ends glued together with the topology of a torus. I had thought that a string with no knots in it is for all practical purposes the same thing, at least for this question. It turns out that they are not the same. In fact, I now can remember learning a few magic tricks that make use of the fact that they are not the same. Thanks to Andy Putman for pointing my mistake out in the comments.

Another update: Thank you to JoshuaZ in the comments for the link to the problem unknot. I tried it out on my own rope and indeed this example shows that the premise of my question is false. Pulling the ends of this rope will not solve this problem.

Another update: Tying shoelaces in double knots is another counterexample that I had forgotten about.

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    $\begingroup$ People have tried similar things, e.g.: arxiv.org/abs/2006.07859 But simulating a physical string means discretizing it and with that it is very easy to accidentally jump in topology. In addition, seeing that this process seems to work in examples is easy. But showing that it always finishes the job in polynomial time seems more or less impossible. $\endgroup$
    – mlk
    Commented Mar 12 at 15:55
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    $\begingroup$ You've missed some key issues. You need your rope to be thick (or else you pull it tight, breaking the topology) and you need your rope to be frictionless. Is that your set-up? If so, you should edit it into your question. If you allow friction you get genuine critical points -- this is the subject of traditional knots, like used in climbing or fisheries. $\endgroup$
    – Ryan Budney
    Commented Mar 12 at 16:10
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    $\begingroup$ Even with a very slippery piece of string, something which is an unknot can seem to get caught when you pull tight on it. For one which I think will show this somewhat, making this unknot out of string and pulling it tight this way after cutting the upper left i.sstatic.net/dtMM9.jpg . $\endgroup$
    – JoshuaZ
    Commented Mar 12 at 17:48
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    $\begingroup$ @CraigFeinstein: No. Here is a picture of an unknot. If you grab it at the red and blue points and pull, it will not come undone. imgur.com/a/lJKSlrj $\endgroup$
    – Andy Putman
    Commented Mar 12 at 19:33
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    $\begingroup$ I think the downvotes are excessive. Yes, the question is vague, but there is a real, research-level question lurking here as evinced by the various references people have provided. $\endgroup$
    – Sam Hopkins
    Commented Mar 12 at 19:40

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Here is a paper that, I think, underlines some of the difficulties in recognising the unknot using the physical process of "pulling tight".

Nontrivial embeddings of polygonal intervals and unknots in 3-space

by Cantarella and Johnston. They prove that there are "stuck" stick unknots for any stick number $n \geq 6$. That is, polygonal knots that are topologically the unknot but cannot be unknotted without changing the lengths of the sticks (or adding breakpoints).

As another piece of negative evidence we have the paper

Topological and physical link theory are distinct

by Coward and Hass. They give a link (made of "rope" - thus each component has a given thickness and length) which is topologically unlinked but cannot be physically unlinked.

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    $\begingroup$ Your question said "since in real life it is quick, at least in my experience". These papers are pointing out that physical experience may be misleading you. Your comment is saying that these counterexamples "might be problematic in the physical world but not in the virtual world". But you've not given me any hints of how you wish to proceed in the virtual world... so how can I evaluate your plan to recognise the unknot? $\endgroup$
    – Sam Nead
    Commented Mar 12 at 22:42
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    $\begingroup$ Your argument is an “intuition pump”- en.wikipedia.org/wiki/Intuition_pump - which is then followed by a “bait and switch” - rationalwiki.org/wiki/Bait-and-switch - … To clarify - I am not suggesting that you are arguing in bad faith… $\endgroup$
    – Sam Nead
    Commented Mar 13 at 8:05
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    $\begingroup$ Instead I am saying that all people have emotional attachment to their positions, which makes it hard to understand other people’s attitudes and backgrounds. $\endgroup$
    – Sam Nead
    Commented Mar 13 at 8:05
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    $\begingroup$ You may be interested in the discussion of physical knots between Tim Gowers and Bill Thurston here: mathoverflow.net/questions/53471/… - in particular they end by agreeing that there are "tangled marionettes" that are very hard to untangle. I regard this as evidence against the claim that "in real life, it is not so difficult to recognize an unknot". $\endgroup$
    – Sam Nead
    Commented Mar 13 at 13:54
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    $\begingroup$ Negative results are still results! Good luck on your journey. :) $\endgroup$
    – Sam Nead
    Commented Mar 14 at 15:58
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As noted in the comments, it is difficult to make this intuitive idea mathematically precise. In practice, even slip knots get stuck on themselves when pulled tight.

One may look for monotonic ways of untying an unknot. Holding a long unknot and pulling on the ends, one may expect for example that the number of maxima with respect to direction between the ends does not increase. Otal proved that one may always undo an unknot without increasing the number of maxima, so there is no obstruction of this type (he works with closed knots, but I think the proof should work for long unknots, ie unknotted strings/1-tangles). Dynnikov proved that one may monotonically simplify a (closed) unknot in grid position /arc diagram, but it’s hard to see how this might translate into something physical by pulling both ends of a long unknot.

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I think your description of a knot as a string with endpoints pulled apart is similar to embedding knots in $\mathbb{RP}^3$, with the free endpoints corresponding to a single point on $\mathbb{R}^3$'s boundary. There are several papers on knots in $\mathbb{RP}^3$. The most notable related knots in $\mathbb{RP}^3$ to knots in $\mathbb{R}^3$ by sliding a small reflecting sphere along the knot and looking at the knot's reflection. IIRC, there are examples of unknots in $\mathbb{R}^3$ that cannot be untied monotonically.

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