What about this sequence?

Zero 011237 2149.

What's the next number?

I'm gonna get a 73.

No, I'm just guessing, Of course on, you know, in the ordinary person would have no clue that seemed like a totally vacuous ears.

And I tell you, it's 1 65 says why on earth at 1 65 this is the number off different mathematical knots.

Visit given crossing number.

Basically, it means there are no mathematical knots.

These one crossing or two crossing, there's one not with three crossings.

There's one, not his four crossings.

The two knots is five crossings.

They're free nuts or six crossings, and then it starts to take off.

And really, we get to a lot of different knots when we have a big number of crossings.

But let's first talk about mathematical knots, and it's not maybe what you typically you think, Is it not?

You see all kind of things that look Oh, yeah, yeah, yeah.

These are definitely not slight.

All kind of really beautiful knots.

And, of course, here my shoe laces form.

But already people would think.

And not now.

Nothing of this is a mathematical, not because it has loose ends.

If it has loose ends, he can in principle, untie it and you can for a different not.

And so there's nothing that you can say about these particular nuts because nothing is top logically locked in.

You can always undo the North and do something differently.

So the ideal, simplest possible mathematical Not maybe just a rubber band.

Okay, well, indeed, mathematicians call this the UN not or the trivial, not because simply not very interesting.

But if you take this rubber band, this is Oh, I cannot it and you try to make some kind of not into it and you stuff It threw itself.

And you two were thing, and it may look very not it.

Well, it's not any different from what you started this because I can go and undo the same thing.

And it's still the same simple ring.

Well, are there any nontrivial mathematical knots that they're not just opening up into a single loop?

And yes, there are, for instance, the simplest.

That knot is the trifle, not it basically a closed loop, as you can see, in this particular projection, we can see it has three crossings.

There's one crossing here.

There's one crossing here, this one crossing there as I change this, not the round.

It may look quite different, and at some point it may have more than just Lee crossings.

But it never has fewer than three crossings.

It may be quite irregular.

For instance, you know, what is this thing?

Certainly has a lot of crossings, but again, we're looking for the minimal number of crossing.

So we try to untangle this and open it up as much as we possibly can.

And if you're patient enough and do that, we can see Oh, look, three crossings has one crossing there.

There's another crossing there.

There's another crossing down there and nothing I can do is get fewer than three crossing.

Of course, I can twist it.

Make 1/4 crossings.

Spend it here.

Make that six or seven crossings.

But they're looking for the minimal number.

And the minimal number in this case happens to be three crossings and with three crossings For a mathematician, these are the same knots.

They're both the trifle not How do you know when you've reached the minimum number of crossings.

Well, you put your finger on a very sore subject because you don't now for simple knots.

Best No question that these are the simplest possibilities.

But say you're going to 2013 crossings or something that that you can never be quite sure.

And as a matter fact, that's still an open problem.

To really have a very complicated Gordian knot and figure out what type off Not today's, but because of that, because once you have the simplest view off, a Not in the way, that's the one that has the fewest crossings that tells you what class you're in.

Now.

If it happens to be three crossings, tenders only one possible, not the train for actually, it's not quite true, because if you were to make a mirror image, you could not be form this into its own mirror image.

But mathematicians say Okay, look, we can mirror this thing so they're not counting that separately.

Now can we make a knot four crossings here?

He's such a nut, and in this particular projection you probably just see four crossings myself.

One over here, there's one over on that side and then this one in here and that one down there.

And it turns out two things.

There's only one type off, not his four crossings.

And contrary to the trifle, this actually is the same as its mirror image.

So I can deform this into its mirror image.

The matter.

Fact.

Let's look at that.

Not in a little bit more detail.

Here is the classical image that you typically see in a table of four knots.

And that's why I guess it's called Figure right now because you can see an obvious figure eight in there.

That's not the only way.

And all you see is all the time.

1234 crossings.

So all of these are top illogical.

The same thing, the one that I showed you before in Vire, would be shown like this.

In this particular diagram, you could see these knots all look quite a bit different, really, their three dimensional and so people have made more three dimensional rendering.

So it's something that's very close to the first classical image that we showed that simple diagram.

So this is your figure eight knot on Deacon.

See, it looks yet quite different again, but in no protection.

Really?

Ever see fewer than four crossings on Dhere?

We are.

This is the real thing, you know.

Solid bronze is a nice green patina done by Steve Ramos in a huge in Oregon.

Absolutely, we'll call it.

It's heavy.

Wow.

Okay, that's a foot spa call that for.

That's a figure eight knot and it's a four crossing, not on.

We have seen the trifle not, and we have seen the figure eight knot.

Now, if you look at the numbers below here, the first number here's a three years before they indicate, What's that minimal number of crossings?

A.

So that's known as the crossing number.

The second number is simply a serial index that tells you this is the first of two possible kinds off knots this five crossings.

They cannot be transformed into one another truly different, and it has been shown that there's no third possibility.

So if you have exactly five crossings as a minimal number, either we have this not or you have that not or they're mirror images.

As we go to six crossings, we now have three possible options and they're all different and they're just serial number 123 In an arbitrary or by the time to go to seven crossings, we have seven different possibilities.

So that's the first classifications off mathematical Nazi.

Simply try to figure out what's the minimal number.

And when people first tried to tabulate these not table.

So some 100 years ago there were some tables.

Originally, for a long time, it was believed.

I think it's the eight crossing knots, that they're more one more.

But then actually two over Overlooked and other store not be duplicates.

So it's really, really challenging.

You would think that maybe a computer program can just take a view of an artery knots and are pretty diagram.

Tell you what, not it is.

No, that's unfortunately not true.

It's true, maybe for the simple knots that have, you know, maybe up to 10 or maybe even 12 crossings.

But then it gets really, really hard because it's so difficult to know how to deform.

You're not to first of all, get to the minimum number of crossings and then how to reshape it in that shape.

So it looks exactly like something that you already have in your table.

People have tried to come up with some mathematical measures to try to.

It's Dr Knots in a formal way.

Absolutely.

And so the crossing number is certainly one of those ways of representing and not another way to represent knots would be that you're trying you flat.

You're not as much as much as you can in the plane.

And like all of these are so 2.5 dimensional diagrams which are playing the diagrams make sickly recess over on the crossings.

So assume you're on the road and you're traveling a wrong one off those paths.

And every time you come to crossing, you start numbering them on the number of air.

It's over and under.

So on the first crossing, you go under an equal number one next trusting.

You may still go on the number two, then you go over number three.

Sometimes you come back to some crossing off already being Maybe you haven't under past.

Now you're on the overpass.

Okay, so at least that again.

But you remember that underpass three is the same as overpass 14 and then overpass four is the same as on the past 22 and that's another way of describing how that knot is really linked.

People like John Horton Conway.

He tries to chop up these north diagrams into smaller pieces that have exactly four strands coming out of it.

And then they have a little bit off naughtiness inside and group them together and link them together and gradually build a complex not out of that.

And that led to another very formal description that can describe any any not you'd like to describe.

Mr.

Jones has come up with some kind of a complicated polynomial that the tribe not is Mr Alexander and has created an Alexander polynomial.

So you think that, you know, is all this.

We really know what's going on.

Unfortunately, we still do not know what's going on.

If I give you really, really complicated knots and you and ask you Are they the same?

First you try to startle, find out what the crossing number is.

Of course, if the crossing number is different, you know they cannot possibly be the same.

Then you struggle this, T.

Jones, Polly, No one really struggle with the Alexander Poley normal.

If any one of these Pelino meals is different, we know that you're not so different and they're too earthly Maur characteristics of knots.

But even if all the known characteristics off those two knots are exactly the same, we can still not be sure that there, indeed to say not so.

There's still an open challenge to find a true.

The script that really will will help to make it quite unique when two knots are different.

So mathematicians still don't fully know what I'm not is that can't fully.

They haven't fully God.

Well, they know what the knot is.

They just don't know if a particular not exactly which label glacier designed to it and very tight to complicate knots are truly different or not.

Thing is not human day, and I know let's assume this is either one of those loops or it's a circular.

The end, a molecule that could be a bacterial genome.

Or it could be just a naturally occurring plasma.

And you are not a knot expert, just a not the file.