This knot is wild.
No, really, that's the technical term for it.
It's like an ordinary knot, except that it's infinitely complicated.
Okay, well, this 3D print is not itself infinitely complicated.
I had to stop at some point, but you get the idea.
This looping pattern should keep repeating, getting smaller
and smaller forever.
But does this really count as a knot?
Are we allowed to tie infinitely many tangles in a piece of string?
Mathematical knots are like knots you would tie in real life, except that
we join the ends of the string together to make a loop so it can't come undone.
Okay, there are some other differences as well.
Mathematical knots don't usually have any thickness,
unlike a real piece of string or a 3D print, it should be infinitely thin.
More precisely, a mathematical knot is a "simple closed curve" in space.
So what's a curve?
A curve is the image of a continuous map of a line segment into space.
For example, this knot, the "trefoil", is the image of the interval
from 0 to 2 pi under this continuous map.
A curve is "closed" if it closes up into a loop like this one does.
And it's "simple" if it doesn't pass directly through itself.
In other words, the map never sends two different points
on the interval to the same point in space.
So here I've collapsed the trefoil down, and now it's no longer
a simple closed curve.
Now, there's nothing in that definition that means you cannot have an infinitely
tangled knot.
You might worry that there's some problem with an infinitely
complicated continuous map, but this turns out not to be a big deal.
Here's a wonderful example of an infinitely wiggly curve
called the "topologist's sine wave".
It's just the graph of the function
y equals sine of one over x, for x greater than zero.
As x gets nearer to zero, one over x gets bigger and bigger, faster and faster.
And we zoom through the wiggles of the sine wave faster and faster.
All the waves kind of bunch up as we get towards x is zero.
Okay, maybe there's a problem with closing this curve up to make a loop,
because it's not so clear
where the left hand end of this curve is when we get to x is zero.
So instead we can just scale the function by multiplying by
x to get y is x times sine of one over x.
And now the function goes to zero as x goes to zero.
And so we can join it up like this to make a closed curve.
And we still have an infinite amount of wiggling happening in this finite
amount of space.
Okay, back to knots.
So a knot is a simple closed curve.
But there's another thing, which is that
we don't care about the exact shape of the curve.
If you deform it a bit, it still counts as the same knot.
And then you might ask, is this really knotted?
Or could you somehow deform it so much that you would untie it, turning it
into the unknot, which is just a circle. And it doesn't seem likely for this knot,
and yeah, it turns out that this knot
really is knotted, although it's not so easy to prove that.
In fact, there are infinitely many knots that are all really different.
Meaning that you cannot deform one into another.
Although again, showing that they're really different
from each other isn't so easy.
All of these knots are "tame",
which means that they can be deformed into polygonal loops.
That is, you can make a version of the knot,
which is made from a finite collection of line segments connected end to end.
For example, this is a polygonal version of the trefoil
knot made from nine line segments.
And the idea is that if you can make a polygonal version of the knot,
then it cannot be infinitely complicated.
It cannot be wild. So instead it's tame.
Here's that knot from the start of the video.
Now these tangles go on forever with smaller and smaller loops,
so it seems like there should be no way
to draw this as a polygon with a finite number of line segments.
But it's not that easy to show that this is wild.
You know, maybe there's some big, complicated way to deform it and somehow
get rid of the infinite complexity before you try to make it out of line segments.
Our scaled topologist's sine wave from before also has infinite complexity in a sense,
even though you can just squish that down into a circle and a circle
is just the unknot which of course can be deformed into a polygon.
So even though this
knot is infinitely wiggly, it's tame because it's just the unknot.
Here's a really interesting example due to a mathematician called Ralph Fox.
You could call it a "wild slipknot".
Just like the other wild knot, it looks like it has infinitely many
tangles, but, well, we can just start undoing each tangle from the end.
If you pull this loop through here,
I mean, it just goes away and the knot looks
the same as when we started, except that it's one loop shorter.
So surely we should be able to just keep doing this,
untangle everything,
and this is just the unknot.
Well "hold on", you might be saying, "this is one of those cases
where infinity does strange things".
It has infinitely many loops
and if you remove one, it still has infinitely many loops.
So to untie this wild slipknot, you'd have to do infinitely many "unlooping" moves.
But is that a problem?
In Zeno's paradox of Achilles and the tortoise, Achilles
and the tortoise are in a race, but the tortoise gets a head start.
Achilles is faster.
But by the time he gets to
where the tortoise was, the tortoise has moved forward a bit.
And by the time Achilles gets to that second point where the tortoise was,
the tortoise has moved on to a third point and so on.
So Achilles has to do infinitely many things
before he can actually overtake the tortoise.
but of course, he can overtake - each step of getting to where the tortoise
was happens faster and faster,
and he overtakes in a finite amount of time,
even though he does infinitely many of these "catching up" moves.
So can we untie the wild slipknot?
We would have to do
infinitely many unlooping moves, but if Achilles can do it, why can't we?
This gets very subtle,
and now we have to be really careful about what it means to deform a knot.
It turns out that with the precise mathematical sense of "deform",
you cannot untie this wild slipknot.
I won't go into the full details, but I can at least wave
in the general direction of a proof that this is not the unknot -
that this isn't the same as a circle.
One of the ways you can
show that a knot is not the unknot is to show that it is "tricolourable".
A knot is tricolourable if you can colour the strands of a diagram of the knot
in three colours, with the rule
that at every crossing you either see all three colours or only one.
And of course, you're not allowed to use only one colour for the whole diagram.
You have to use at least two.
This diagram of the trefoil knot is tricolourable,
and if you start moving the knot around to change the diagram, doing
"Reidemeister moves" as they're called, it stays tricolourable.
But this diagram of the unknot is not tricolourable.
There's only one strand to colour in, and we have to use at least two colours.
So the unknot is not tricolourable,
the trefoil is tricolourable, so they must actually be different knots.
And it turns out
that our wild slipknot can be tricoloured.
Here's how you do it.
Well, maybe this is not such a satisfying argument
for why the wild slipknot cannot be untied.
The usual justification for why tricolourability is an invariant of the
knot is that if you do any finite sequence of these Reidemeister moves,
you either stay tricolourable or you stay not tricolourable.
But no finite number of moves is going to undo the wild slipknot anyway.
And it isn't so clear how to keep track of tricolourability
when you do some infinite sequence of moves.
But there's another, more technical way to think about tricolourability.
Rather than colouring the strands, you think about the loops
that go around the strands back to some fixed base point.
You can "multiply" loops together just by doing them one after another.
And it turns out that these loops form
a group called the "fundamental group of the complement of the knot".
The fundamental group is calculated in terms of the topology
of the complement, which means that it doesn't depend on the diagram.
So there's no argument involving finitely many Reidemeister moves.
To show that this, wild slipknot is not the unknot,
all we have to do is show that its fundamental
group is not the fundamental group of the unknot.
That is, we have to show that it is not the group of the integers.
You can do this by showing that there is a non-trivial homomorphism
to a noncommutative group, because the integers form a commutative group.
For us, that group will be the simplest possible noncommutative group:
the group of symmetries of an equilateral triangle.
The relations among the loops in the fundamental group
come from the crossings of the knot.
If you go around the over strand, then one of the strands, then
the over strand in the other direction, and then the other under strand,
then you really didn't go around anything.
This is one of the
relations of the so-called "Wirtinger presentation".
So let's do that same relation with the symmetries of the triangle.
Let's send the three colours of our tricolouring
to the three reflection axes like this.
First we flip across the green axis, then the blue axis,
then green again, and then finally red.
And we end up back where we started.
So the relations of the fundamental group also hold when mapped to the symmetries
of the triangle.
And we have a non-trivial homomorphism to those symmetries.
So that means that the group we started with could not have been commutative,
So it could not have been the integers,
and so the wild slipknot could not have been the unknot.
It isn't the unknot.
Okay. That's it.
Thanks to my collaborator Saul Schleimer for working this out with me.
And thanks to you for watching.