Usbrth

Somewhat surprisingly, I don't see a question about this.

There is a team-building (or just fun mathematical) game where a group of people hold hands with each other, usually trying not to hold hands with someone right next to you. The goal is then to "untangle the human knot" thus formed.

Folk wisdom says this can always be "solved", in the sense of twisting and moving to demonstrate the knot formed is just one unknot, but this isn't so, since you can form all sorts of knots. Probably forming a simple non-trivial link of circles would be easiest to demonstrate this.

But I am intrigued by a lack of easy-to-find references on the probability of such a configuration being the unknot. There is this MathOverflow question, which however has devolved into whether any link can be formed, which is NOT what I am asking. See also this Reddit thread and this Quora thread.

In any event, not only do I feel like probably there is a known answer, it is also not possible to search on this site for questions on MO, so hopefully it is appropriate to ask on MSE this question:

Given any reasonable set of definitions of this game and reasonable probability distribution given your definitions, what is the probability that such a link is the unknot, as a function of $n$ players?

Presumably this will vary by some assumptions on arm length or the exact rules (can you grasp your neighbor's hand, how are people arranged), so there could be multiple answers. I suppose it's likely the parity of $n$ will be involved as well.

As a hint, there is a comment to the MO question suggesting some possible references in somewhat difficult-to-access resources - but I don't care about references per se, I would like answers that are publicly available on a user-friendly and well-indexed site ... such as this one!

In a somewhat related MathOverflow question, where a closed loop is chosen at random as a polygonal path whose vertices lie on a sphere, there are some thoughts that the average crossing number of such a knot is something like $n^3/2$, where $n$ is the number of vertices. This would mean nontrivial knots are reasonably likely as the number of people grows, especially since the maximum crossing number of any such knot is bounded above by $n^2$.

Even-Zohar has a paper on models for random knots. The random jump model is sort of like the human knot, but people are allowed to be placed anywhere in a unit sphere. Numerical experiments suggest the probability of encountering an unknot vanishes faster than $exp(−O(n))$.

In that paper, there is a description of a model that is much closer to the human knot game: random grid diagrams. If I understand it correctly, the difference is that the order in which people hold hands matters. Figure 15 has a graph showing the sampled distributions for the Casson invariant $c_2$ (the order-$2$ Vassiliev invariant, the second coefficient of the Alexander-Conway polynomial) of random knots from different models (including the grid model). The value of $c_2$ for the unknot is $0$. If I'm reading the graph correctly, with about eighty people the probability of getting an unknot happens no more than $55%$ of the time. The actual probability is less since other knots also have $c_2=0$.

The paper cites what appears to be a 2007 PhD thesis by Gilad Cohen in which the human knot game is numerically analyzed. However, I cannot find a copy or a reference to it anywhere.

In one experimental analysis of the grid diagram model, they find the knotting probability approaches $1$ as $n$ increases. As an example, it looks like the human knot game (conditioned on people always forming a single closed loop) for ten people has nearly a $20%$ chance of being unable to be detangled, though I can't say for certain since I don't know how well this random model actually maps to the human knot game.

Anyway, the short of it is that it appears the answer is unknown right now, but there is numerical evidence to support the conjecture that, as the number of people playing the game increases, winning becomes arbitrarily improbable.