Name: What is a Knot? - Numberphile
Uploaded: 2020-03-30T06:45:55.000Z
Duration: 10 min 52 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

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.

Basically, it means there are no mathematical knots.

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

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.

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, 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.

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 down there and nothing I can do is get fewer than three crossing.

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.

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.

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.

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.

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.

Ever see fewer than four crossings on Dhere?

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

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?

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.

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What is a Knot? - Numberphile

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