I recently played the human knot gamewith my children and family. It’s real fun!! We untied one knot with four of us but couldn’t solve one when six were involved.
Is it possible to solve ALL human knots? If not is there some way to predict what proportion of knots in a circle with a given number of participants are solvable?
This site states that some knots are unsolvable, but I doubt they that is the final word on the subject.
I have been in a few knot games that ended up with two linked circles, two seperate circles, or some kind of weird tangle that we couldn’t figure out. These were big games, though, with like 10-12 people, which probably increases the number of od formations you can come up with.
I think that this guy pretty much proved that they all have solutions. 
You can get a three-person knot that’s unsolvable. Here’s how:
[ol]
[li]Person A joins his left hand with Person B’s right hand.[/li][li]Person C reaches her right hand over Person A’s left arm. Person B takes Person C’s right hand with her left hand.[/li][li]Person C reaches her left hand under Person B’s right arm. Person A reaches his right hand over Person B’s left arm. They join hands.[/li][/ol]
If I’ve described this correctly, their hands & bodies (topologically speaking) form something called the trefoil knot, which cannot be “untied” without the people letting go of each other’s hands. From here, it’s not too hard to see that any number of people can form a knot like this — so it’s possible for any number of people to be “un-untanglable.”
When I was a kid, we called this game “Dr. Tangle.” The way we played it, one kid was chosen to be the doctor and was sent out of the room. The rest of the kids would join hands in a circle, and would then tangle themselves while still holding hands. When we decided we were sufficiently tangled, we would call for Dr. Tangle, who would come in and try to untangle us. In this version of the game, all knots are untangleable.
It’s pretty each to come up with unsolvable knots if people are allowed to join hands any way they want. Suppose there are four people. Two of the people opposite each other join hands by reaching across the circle, with each person’s right hand holding the left hand of the other person. Now the other two people reach across the already-joined arms and join hands the same way, except that each person puts his right arm over the arms closest to him, and his left arm under the arms closest to him. This forms the equivalent of two joined rings.
Basically, if all you want to do is grab hands arbitrarily and get into one unknotted circle then yes: there are a huge number of possible links. In fact, some part of my job is to figure out when you can and can’t do this.
Dror Bar-Natan’s table of prime knots up to ten crossings.
Now, if you start adding restrictions it becomes an interesting question how many links are possible, and with what “probability” (in some suitably-defined sense). The “not your neighbor” restriction is largely to make a touchy-feely political point, so I’ll omit that for now. More importantly, we have a finite number of people. We can probably assume they aren’t going to try to tie a knot in their body, but rather just reach straight to the other person’s hand. How they over- or under-cross other arms is open, so we’ll assume arms are bendy enough that any over/undercrossing is possible and stretchy enough to reach anyone else in the circle. Now it becomes a question of a knotted embedding of an n-gon. Possibly tightening the restrictions and weighting the probability of grabbing a near-neighbor’s hand rather than across the circle will make an unknot more likely, but a good combinatorial model should come first.
As for whether anyone’s worked this out, I’m not sure offhand. I’d start with Freyd and Yetter’s work on knotted graphs, particularly the article of one of them (I forget which) in Advances in Mathematics 77 (1989). And, in case you’re wondering, yes those are the F and Y from HOMFLY.
Or of multiple polygons whose numbers add up to n. With six players, for instance, you could join hands in a Star of David, with two unconnected triangles (which may or may not be linked, of course).
And I think that the “not your neighbor” restriction is meant more to make it harder to untie, and therefore more fun, than it is to make a political point. That government resource page which says the game teaches ecology is a real stretch, and I suspect that most people who’ve played this game have never heard of that rationale.
In general it’s to make some sort of “teamwork” point, adapted politically on that page.
You’re right about the multiple polygons, of course. I think I may have been confounding this presentation with the one I saw as a child. That one actually does manage to always result in a single component link, and an unknotted one at that.
Start in a circle as before. Now one person grabs another’s left hand with their right (exercise: this loses no generality). That person grabs anyone else’s free left hand but the first, passing their arm over the first arm if necessary. Continue, each new person grabbing another person’s free left hand other than the first person’s (unless it’s the only one remaining), passing the new connection over any existing connections. The resulting knot cannot have more than one component, and that component is ambient-isotopic to the unknot. The first statement is an easy exercise, and the unknottedness shouldn’t be that hard. If needed, refer to Crowell & Fox’s Introduction to Knot Theory for the proof.
Is there a general way I can visualize converting human knots into those depicted in your link? I think the number of crossing is equivalent to the number of people involved.
The link includes “prime” knots. How is this defined?
Also, which of the knots in the link are solvable in the human knot sense?
The crossing number has almost nothing to do with the number of people. Really, the only easy conversion goes from humans to abstract knots: imagine drawing a line along the center of each person’s arms and through their torso, connecting where two people grab hands. Then remove the people. What you have left is an abstract knot as shows up on the table.
If you imagine two abstract knots set beside each other, you can cut each one and splice the new ends together to form one larger knot. This is the “connected sum” operation. For instance the square and granny knots are each the connected sum of two trefoils (3[sub]1[/sub] in the table). Prime knots are those which cannot be expressed as the connected sum of two other nontrivial knots, and there is a sort of “unique factorization” theorem for knots just as for numbers.
As for which are “solvable”: that’s the point of the table. No two of the knots on the table can be deformed into each other without cutting the string (two humans letting go their hands). Thus the only knot that can be “solved” is the unknot itself, (0[sub]1[/sub]).
Mind you, this is only a list up to 10 crossings and omits prime links (knots with more than one loop). There are really infinitely many non-isotopic prime knots.
None of those knots in Mathochist’s link are “solvable”. Any solvable knot is topologically the un-knot; that is to say, just a simple loop of rope. The number of people needed is more complicated, since it’s not a mathematical question: In principle, if people’s arms were sufficiently flexible, any number of people could form any knot (well, you’d need at least three people to satisfy the “not both to the same person” requirement, and at least five to satisfy the “not your neighbors” one, but five people with tentacle-arms could make any knot at all). Of course, human arms are not tentacles, so there’s a limit to how much you can tangle them, but I don’t think there’s any straightforward way to express this limit.
Ooh… Ooh… I know the M!
It’s possible to know all from HOMFLY, and all but one from LYMPHTOFU. 