Wild knots

0:00 This knot is wild.

0:03 No, really, that's the technical term for it.

0:05 It's like an ordinary knot, except that it's infinitely complicated.

0:09 Okay, well, this 3D print is not itself infinitely complicated.

0:13 I had to stop at some point, but you get the idea.

0:15 This looping pattern should keep repeating, getting smaller and smaller forever.

0:21 But does this really count as a knot?

0:23 Are we allowed to tie infinitely many tangles in a piece of string?

0:28 Mathematical knots are like knots you would tie in real life,

0:30 except that we join the ends of the string

0:32 together to make a loop so it can't come undone.

0:34 Okay, there are some other differences as well.

0:37 Mathematical knots don't usually have any thickness,

0:39 unlike a real piece of string or a 3D print, it should be infinitely thin.

0:43 More precisely, a mathematical knot is a "simple closed curve" in space.

0:48 So what's a curve?

0:49 A curve is the image of a continuous map of a line segment into space.

0:53 For example, this knot, the "trefoil",

0:55 is the image of the interval from 0 to 2 pi under this continuous map.

1:01 A curve is "closed" if it closes up into a loop like this one does.

1:05 And it's "simple" if it doesn't pass directly through itself.

1:08 In other words, the map never sends two different

1:10 points on the interval to the same point in space.

1:12 So here I've collapsed the trefoil down,

1:15 and now it's no longer a simple closed curve.

1:18 Now, there's nothing in that definition that means

1:20 you cannot have an infinitely tangled knot.

1:23 You might worry that there's some

1:24 problem with an infinitely complicated continuous map,

1:26 but this turns out not to be a big deal.

1:29 Here's a wonderful example of an infinitely

1:31 wiggly curve called the "topologist's sine wave".

1:34 It's just the graph of the function y equals sine of one over x,

1:38 for x greater than zero.

1:40 As x gets nearer to zero, one over x gets bigger and bigger, faster and faster.

1:45 And we zoom through the wiggles of the sine wave faster and faster.

1:48 All the waves kind of bunch up as we get towards x is zero.

1:52 Okay, maybe there's a problem with closing this curve up to make a loop,

1:56 because it's not so clear where the left hand end

1:58 of this curve is when we get to x is zero.

2:01 So instead we can just scale the function by multiplying by x

2:04 to get y is x times sine of one over x.

2:08 And now the function goes to zero as x goes to zero.

2:11 And so we can join it up like this to make a closed curve.

2:14 And we still have an infinite amount

2:16 of wiggling happening in this finite amount of space.

2:20 Okay, back to knots.

2:21 So a knot is a simple closed curve.

2:23 But there's another thing,

2:24 which is that we don't care about the exact shape of the curve.

2:27 If you deform it a bit, it still counts as the same knot.

2:30 And then you might ask, is this really knotted?

2:33 Or could you somehow deform it so much that you would untie it,

2:37 turning it into the unknot, which is just a circle.

2:40 And it doesn't seem likely for this knot, and yeah,

2:43 it turns out that this knot really is knotted,

2:45 although it's not so easy to prove that.

2:48 In fact, there are infinitely many knots that are all really different.

2:51 Meaning that you cannot deform one into another.

2:53 Although again, showing that they're really

2:55 different from each other isn't so easy.

2:58 All of these knots are "tame",

3:00 which means that they can be deformed into polygonal loops.

3:03 That is, you can make a version of the knot,

3:05 which is made from a finite collection of line segments connected end to end.

3:09 For example, this is a polygonal version

3:10 of the trefoil knot made from nine line segments.

3:14 And the idea is that if you can make a polygonal version of the knot,

3:17 then it cannot be infinitely complicated.

3:20 It cannot be wild.

3:21 So instead it's tame.

3:23 Here's that knot from the start of the video.

3:25 Now these tangles go on forever with smaller and smaller loops,

3:28 so it seems like there should be no way to draw

3:30 this as a polygon with a finite number of line segments.

3:33 But it's not that easy to show that this is wild.

3:36 You know, maybe there's some big,

3:37 complicated way to deform it and somehow get rid of the infinite

3:40 complexity before you try to make it out of line segments.

3:44 Our scaled topologist's sine wave from before

3:46 also has infinite complexity in a sense,

3:49 even though you can just squish that down into a circle and a circle

3:53 is just the unknot which of course can be deformed into a polygon.

3:56 So even though this knot is infinitely wiggly,

3:59 it's tame because it's just the unknot.

4:03 Here's a really interesting example due to a mathematician called Ralph Fox.

4:06 You could call it a "wild slipknot".

4:09 Just like the other wild knot, it looks like it has infinitely many tangles,

4:12 but, well, we can just start undoing each tangle from the end.

4:16 If you pull this loop through here, I mean,

4:19 it just goes away and the knot looks the same as when we started,

4:22 except that it's one loop shorter.

4:23 So surely we should be able to just keep doing this, untangle everything,

4:27 and this is just the unknot.

4:29 Well "hold on", you might be saying,

4:30 "this is one of those cases where infinity does strange things".

4:33 It has infinitely many loops and if you remove one,

4:36 it still has infinitely many loops.

4:38 So to untie this wild slipknot,

4:40 you'd have to do infinitely many "unlooping" moves.

4:43 But is that a problem?

4:45 In Zeno's paradox of Achilles and the tortoise,

4:48 Achilles and the tortoise are in a race, but the tortoise gets a head start.

4:52 Achilles is faster.

4:53 But by the time he gets to where the tortoise was,

4:55 the tortoise has moved forward a bit.

4:57 And by the time Achilles gets to that second point where the tortoise was,

5:00 the tortoise has moved on to a third point and so on.

5:03 So Achilles has to do infinitely many

5:06 things before he can actually overtake the tortoise.

5:08 but of course, he can overtake- each step of getting

5:11 to where the tortoise was happens faster and faster,

5:13 and he overtakes in a finite amount of time,

5:15 even though he does infinitely many of these "catching up" moves.

5:19 So can we untie the wild slipknot?

5:21 We would have to do infinitely many unlooping moves,

5:23 but if Achilles can do it, why can't we?

5:27 This gets very subtle,

5:28 and now we have to be really careful about what it means to deform a knot.

5:32 It turns out that with the precise mathematical sense of "deform",

5:35 you cannot untie this wild slipknot.

5:38 I won't go into the full details,

5:40 but I can at least wave in the general direction of a proof

5:42 that this is not the unknot- that this isn't the same as a circle.

5:47 One of the ways you can show that a knot

5:49 is not the unknot is to show that it is "tricolourable".

5:52 A knot is tricolourable if you can colour the strands

5:54 of a diagram of the knot in three colours,

5:57 with the rule that at every crossing you

5:59 either see all three colours or only one.

6:02 And of course, you're not allowed to use only one colour for the whole diagram.

6:05 You have to use at least two.

6:07 This diagram of the trefoil knot is tricolourable,

6:10 and if you start moving the knot around to change the diagram,

6:14 doing "Reidemeister moves" as they're called, it stays tricolourable.

6:18 But this diagram of the unknot is not tricolourable.

6:21 There's only one strand to colour in, and we have to use at least two colours.

6:24 So the unknot is not tricolourable, the trefoil is tricolourable,

6:28 so they must actually be different knots.

6:31 And it turns out that our wild slipknot can be tricoloured.

6:35 Here's how you do it.

6:37 Well, maybe this is not such a satisfying

6:40 argument for why the wild slipknot cannot be untied.

6:43 The usual justification for why tricolourability is an invariant of the knot

6:48 is that if you do any finite sequence of these Reidemeister moves,

6:51 you either stay tricolourable or you stay not tricolourable.

6:55 But no finite number of moves is going to undo the wild slipknot anyway.

6:59 And it isn't so clear how to keep track

7:02 of tricolourability when you do some infinite sequence of moves.

7:05 But there's another, more technical way to think about tricolourability.

7:08 Rather than colouring the strands, you think about the loops that go around

7:12 the strands back to some fixed base point.

7:14 You can "multiply" loops together just by doing them one after another.

7:18 And it turns out that these loops form a group

7:20 called the "fundamental group of the complement of the knot".

7:24 The fundamental group is calculated in terms of the topology of the complement,

7:27 which means that it doesn't depend on the diagram.

7:29 So there's no argument involving finitely many Reidemeister moves.

7:34 To show that this, wild slipknot is not the unknot,

7:36 all we have to do is show that its

7:38 fundamental group is not the fundamental group of the unknot.

7:41 That is, we have to show that it is not the group of the integers.

7:44 You can do this by showing that there

7:47 is a non-trivial homomorphism to a noncommutative group,

7:49 because the integers form a commutative group.

7:52 For us, that group will be the simplest possible noncommutative group:

7:55 the group of symmetries of an equilateral triangle.

7:59 The relations among the loops in the fundamental

8:00 group come from the crossings of the knot.

8:02 If you go around the over strand, then one of the strands,

8:05 then the over strand in the other direction,

8:07 and then the other under strand, then you really didn't go around anything.

8:12 This is one of the relations of the so-called "Wirtinger presentation".

8:16 So let's do that same relation with the symmetries of the triangle.

8:20 Let's send the three colours of our tricolouring

8:23 to the three reflection axes like this.

8:26 First we flip across the green axis,

8:28 then the blue axis, then green again, and then finally red.

8:33 And we end up back where we started.

8:35 So the relations of the fundamental group also

8:37 hold when mapped to the symmetries of the triangle.

8:40 And we have a non-trivial homomorphism to those symmetries.

8:43 So that means that the group we started with could not have been commutative,

8:46 So it could not have been the integers,

8:48 and so the wild slipknot could not have been the unknot.

8:51 It isn't the unknot.

8:53 Okay.

8:54 That's it.

8:55 Thanks to my collaborator Saul Schleimer for working this out with me.

8:58 And thanks to you for watching.