Using topology to close a rubber band bracelet

0:00 Hi, I'm Henry Segerman.

0:01 I'm Sabetta Matsumoto.

0:02 And I'm Saul Schleimer.

0:04 Go maths!

0:05 If you've ever played with rainbow looms or hair ties or rubber bands,

0:10 you may have run into this problem.

0:13 You've made a chain of loops and now you'd

0:14 like to join the ends together to make a bracelet.

0:21 But how?

0:23 Of course you could use an S clip, or a carabiner,

0:27 or you could cut one of the bands and use it to tie the knot.

0:30 but since we are topologists these are all sort of cheating.

0:34 Here's the problem.

0:35 We want to make a bracelet that stays on your arm and does not fall off.

0:39 To be a bracelet, it must pass what we are calling "the bracelet test".

0:43 I put the bracelet on my arm and I clasp my hands.

0:49 If someone else can take it away, then it fails the bracelet test.

0:52 Let's find out if this passes the bracelet test.

0:57 Nope, not a bracelet.

1:00 Here are some more non-bracelets.

1:09 Ok, so the chain itself needs to form a loop.

1:14 Here's something that passes the bracelet test.

1:16 There's no way to take this bracelet off.

1:19 Showing that something cannot be done is often

1:21 harder than showing that something can be done.

1:24 You have to somehow check that none

1:25 of the infinitely many things you might try works.

1:28 We will say more about proving this kind of thing at the end of the video,

1:32 but this chain is sort of cheating anyway.

1:34 The bracelet is made of loops, but you cannot make this bracelet out

1:37 of hair ties without cutting and rejoining.

1:40 Okay, how about this?

1:43 So this seems to pass the bracelet test.

1:45 Again this needs a proof.

1:47 New rule: no cutting is allowed.

1:52 That is definitely cutting.

1:57 NO.

1:59 Here's another bracelet that passes the bracelet test.

2:03 Incidentally, these two ties together with Saul's arms form the Borromean rings.

2:08 There's a problem with the Borromean Rings bracelet, though.

2:11 It's a bit tight.

2:12 It is exactly as tight as a single hair

2:14 tie since each hair tie doubles back on itself.

2:18 So new, harder problem.

2:19 Can you make a bracelet out of very small loops like these?

2:24 This question was asked by Anton Petrunin on MathOverflow.

2:28 He asked: "My suitcase broke, but I have a ton of hair ties." "Can I

2:32 make sure my suitcase stays closed using only hair ties?" Obviously,

2:36 the one or two loop bracelets we saw

2:38 before wouldn't work to go around a suitcase.

2:41 this feels like a knot theory question, but the length of the hair ties compared

2:44 with the size of the suitcase makes it not quite topological.

2:49 If you want to think about this for yourself,

2:51 now's the time to pause the video and let

2:53 us know your solution in the comments below.

2:57 Here's another way to think about it.

2:59 Whatever shape you make with your loops,

3:01 it must hold together when your arm is there,

3:03 but completely fall apart when your arm isn't there.

3:07 Because without your arm there you can just reverse the video of you

3:10 making the bracelet back to when they were just individual separate hair ties.

3:16 We tried to prove that it's impossible to make

3:18 a bigger bracelet for a couple of weeks before giving up.

3:21 And that was a good thing that we did,

3:23 because there are solutions discovered by Michael Freedman and Larsen Linov.

3:27 Here's what you do.

3:28 First, make a chain.

3:29 For example, using fish tails or square knots.

3:32 The chain needs to be long enough to go twice round your arm.

3:37 Next make a slip knot on one end and then make a slip knot on the other end.

3:47 Enlarge the circles until they meet in the middle.

3:53 Now put your arm through both loops and this passes the bracelet test.

4:02 It seems pretty secure.

4:04 But we should prove that it passes the bracelet test.

4:07 We'll actually prove a stronger result: that even with infinitely stretchy,

4:11 topological bands, you cannot remove this bracelet.

4:15 The first step of the proof is to transform

4:17 the person with clasped hands into a topologically identical object.

4:20 A hair-tie.

4:22 It's a different color, red, to signify its special status.

4:26 Now the person has been turned into a hair tie.

4:28 We can treat them more like the other hair ties.

4:30 In fact, they're the same as all of the other hair ties.

4:35 By the way, we made this prop by cutting then rejoining the red hair tie.

4:39 We used a candle to melt it back together.

4:42 This is definitely against the topological rules.

4:45 Unfortunately, to really understand the rest of the proof requires

4:48 a graduate course in geometric topology plus a bit more.

4:52 You have been warned!

4:54 Here in the appendix, we'd like to explain why this can't be

4:58 taken apart into its component hair ties without cutting.

5:03 maybe this one is the human and the others are nicely symmetrically arranged.

5:07 And you see that once the human has been turned into a hair tie,

5:10 there's an extra symmetry which wasn't visible before.

5:12 There's this rotation symmetry around the center.

5:14 So I'm going to call this thing a link, a link of unknots,

5:18 and we need to prove that it can't be moved into the unlink,

5:22 it can't be separated.

5:23 You can't pull the pieces off without cutting.

5:25 Okay, so that's a theorem: "The link L_n" "is not the unlink." Which is

5:34 a fancy name for the pile of hair ties, all separated from each other.

5:37 In fact, we're going to prove more: "The complement" Let's call that X_n,

5:42 which is S^3 minus L_n "is hyperbolic." So

5:47 the complement of the unlink is very much not hyperbolic.

5:49 So this second sentence is sort of a stronger version of the first sentence.

5:53 So the first thing to do is to augment,

5:56 We "augment" L_n by the symmetry axis, let me call that K_n, to obtain,

6:05 let me call it L_n prime So let me draw a picture

6:09 of this and what I'm going to do is remember the symmetry axis is around here,

6:16 so I'm going to take this thing which is coming kind of out and around,

6:20 and I'm going to bend it off a little bit off to the side so that we can see it.

6:24 There's a picture of K_n.

6:26 Let me label it K_n and remember that the white stuff,

6:29 the original stuff was L_n so the union of the two together is L_n prime.

6:34 We also need to define X_n prime equal to S^3 minus L_n prime.

6:41 So the complement, the thing that the knot is not,

6:44 that gets smaller as the link gets bigger.

6:48 So we had the link L_n we added something to it.

6:51 That's the same thing as drilling out a curve

6:54 from X_n and that gives us X_n prime.

6:57 So X_n prime is obtained from X_n

7:02 by drilling So this link with the added component,

7:06 it still has the symmetry and it has something extra you can do.

7:10 You can take covers by unwrapping copies of L_n and K_n just becomes,

7:16 in some sense, it's a bigger unknot Note

7:22 that X_{l n} prime covers, l-fold, X_n prime.

7:29 In particular we have that X_n prime covers, n-fold, X_1 prime.

7:38 Recall that X_1 prime equals S^3 minus L_1 prime.

7:43 So I have a picture of that here: If we only had one link in our chain,

7:49 then it would be a single hair tie which is kind of clasping itself.

7:54 The white stuff is the hair tie, but we still have the augmentation,

7:58 which is now called K_1, because there's only one link in our chain.

8:01 So the important point about this is the following:

8:05 We use a computer program, namely Snappy.

8:08 "X_1 prime is hyperbolic" "and the meridian of K_1"

8:15 "has length exactly five." So what's the meridian?

8:18 The meridian is the curve, which is going around here this kind of dual to K_1

8:23 and it's sort of measuring how far around the manifold goes.

8:28 And when I take a cover the meridian

8:31 of K_n in X_n prime has length five times n.

8:37 And maybe the important thing here is that if

8:40 n is large then 5n is five times larger.

8:42 so now we're going to state the two

8:44 pi theorem which is due to Gromov and Thurston,

8:49 "If you fill along a curve of length" "greater than or equal to two pi,

8:59 then" "the result is pinched negatively curved."

9:03 So maybe I should back up a step.

9:05 I claimed that this manifold X_1 prime

9:09 is hyperbolic and the meridian has a certain

9:11 length and I said we're going to use a computer program to verify that.

9:15 So this can be done with interval arithmetic.

9:17 So that's...

9:18 that's a theorem.

9:20 And then I said "Okay, there's this two pi theorem of Gromov" "and Thurston".

9:23 And what we're doing is, if you fill along a curve, so filling a curve,

9:29 adding something to the complement means just taking this and erasing it.

9:34 Right, so if we went back to the link L_n prime and deleted K_n,

9:40 that's the same thing as filling So what we learn from all this is:

9:46 Thus X_n admits a metric of negative curvature.

9:50 So then by a result of Thurston X_n is hyperbolic

9:55 (If five times n is greater than two pi That is,

10:01 if n is greater than or equal to two.) Finally X_1 is not hyperbolic.

10:07 And that's because if you look at L_1 all by itself,

10:12 you can see that it is a connect sum of a pair of trefoils.

10:17 So that's not hyperbolic.

10:19 So this is the final result.

10:21 We get exactly what we needed that all of the links L_n,

10:26 when n is greater than or equal to two, are hyperbolic so they can't come apart.

10:31 And actually the link L_1 it also can't come apart because, well,

10:34 it's only got one component, but it's,

10:36 it's a connect sum of trefoils- it's not the unknot.

10:39 There are a few other links which we need to prove are nontrivial.

10:42 the first of these is the Hopf link.

10:44 this comes about when we have one of them being

10:46 a hair tie and the other one being the human.

10:49 One proof that this is non-trivial uses linking numbers.

10:52 Which is due to Gauss.

10:54 The next link we need to prove is non-trivial is the Borromean rings.

10:57 And here one of the components was the human,

11:02 and the other two were the bracelet.

11:05 So you can prove this is not trivial using hyperbolic geometry.

11:07 That's a quite different proof from the one before,

11:10 or using sort of a higher order linking numbers called Milnor invariants

11:14 due to Milnor And I think the last example, is the metal chain.

11:19 So the big component here is the human, and then these are linked to each other.

11:24 So you could use hyperbolic geometry very similar to the previous discussion,

11:28 or you could use linking numbers to prove that this is non-trivial.

11:32 Thanks for watching!