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  • Translator: Reiko Bovee Reviewer: Ivana Korom

  • Theoretical physics.

  • What does that make you think of?

  • (Laughter)

  • Maybe you had physics in school,

  • or maybe you think of one of the greats like Albert Einstein.

  • Maybe you think of fundamental particles:

  • the elementary building blocks of our universe.

  • I am a theoretical physicist, and I think of these things,

  • but I spend an awful lot of time thinking about knots.

  • What I usually want to know about knots

  • is whether one knot is the same or different from another knot.

  • What I mean by this is: can the knot on the right be twisted

  • and turned around and turned into the knot on the left

  • without cutting without using scissors?

  • If you can do this, we say they are equivalent knots,

  • otherwise we say they're inequivalent.

  • Surprisingly enough, this question of equivalence of knots

  • is very important for certain types of fundamental particles.

  • Furthermore, it's important for the future of technology.

  • This is what I am going to tell you in the next 15 minutes.

  • To get started we need some of the results from relativity.

  • Now relativity is a pretty complicated subject,

  • I am not going to explain much of it.

  • One of the themes we learn from it

  • is that space and time are mostly the same thing.

  • So, I've a little story to explain this,

  • it's a story of Einstein's world and his day.

  • So, we have his home, his work, the cinema on the screen,

  • and there's a clock in the upper right hand corner,

  • so keep your eye on the clock during the day.

  • Eistein starts his day, and he goes to work,

  • then after a while, he comes home for lunch, the clock keeps ticking,

  • he goes back to work, the clock keeps ticking,

  • in the afternoon he decides to go to cinema, he goes to the cinema,

  • the clock keeps ticking, and eventually he goes home.

  • Physicists would look at this and want to treat time more similarly as space,

  • and the way we do this is we plot space on an axis,

  • and we plot time on another axis.

  • Eistein's so called "World Line" is this dark red line

  • which tells you where in space he is at any given time.

  • It's called World Line because it tells you

  • where in the world he is at any given time.

  • Now we can go through the day, keep your eye on the dark red ball.

  • The ball goes up one step every hour as we go through the day.

  • It goes back and forth in space, tracing Einstein's position.

  • So the world line is just a convenient way of keeping track

  • of where Einstein is at any point in the day.

  • We can do the same thing with a more complicated world.

  • So here we imagine looking down on Einstein's neighborhood

  • from a helicopter above.

  • Einstein starts his day at home,

  • he goes to work, he goes to the cinema, he goes back home.

  • A student on the same day starts at home,

  • goes to the library, goes to the pub and goes home.

  • If we follow them both on the same day,

  • Einstein goes to work, the student goes to the library,

  • Einstein goes to the cinema, the student goes to the pub,

  • Einstein goes home, the student goes home, it starts to look pretty complicated.

  • But we can simplify it

  • by looking at the space-time diagram of what happened.

  • We do that by turning the neighborhood sideways,

  • plotting time vertically and notice I've drawn a blue vertical line

  • at the position of every object in the neighborhood that doesn't move,

  • such as the library and the pub;

  • they stay fixed in space and move through time.

  • Einstein and student's world lines move around in the neighborhood

  • as they go through time.

  • Now we can kind of see where I am going with this.

  • Einstein and the student's world lines have wrapped around with each other.

  • If you pull those tight, you'll discover you have it knotted.

  • We need one more thing from the theory of relativity.

  • We need E = mc².

  • Again, this is a thing that I am not going to explain to you in much depth.

  • Roughly what it means is that energy and mass are the same thing.

  • If we have a particle in the world like an electron -

  • that's a particle of matter.

  • Each particle of matter has an opposite particle of antimatter.

  • In the case of electron,

  • the antimatter particle is called the positron.

  • Both the electron and the positron have mass.

  • If you bring them together, however, they can annihilate each other,

  • giving off their mass as energy, usually as light energy.

  • The process works in reverse just as well.

  • You can put in the energy and get out the mass of the particles.

  • Now, we are going to do the same thing we did with Einstein's neighborhood.

  • We are looking down on the neighborhood,

  • we put in energy to create particle and antiparticle.

  • We put in energy to create particle and antiparticle.

  • Then we move one particle around another, and we bring them back together,

  • we reannihilate them, we annihilate them; releasing the energy again.

  • Now if we'd look at that in the space-time diagram,

  • it looks a little bit like this;

  • time running vertically, we put in the energy, we put in the energy

  • we wrap one particle around the other,

  • and we annihilate them, and annihilate them again.

  • We can see quite clearly here

  • that the world lines have knotted around with each other.

  • We did the same thing with more particles, by putting in more energy,

  • move them around in more complicated way and bring them back together.

  • The space-time diagram would look a little bit like this,

  • making a very complicated knot.

  • Here's the amazing fact upon which the rest of my talk relies.

  • Certain particles, called Anyons exist in 2+1 dimensions.

  • Now I should probably say what I mean by 2+1 dimensions.

  • Two dimensions mean we're talking about a flat surface,

  • so these particles live on flat surfaces.

  • We say +1 dimensions, we mean also time.

  • We are just saying that particles on the flat surfaces move around in time.

  • So, these particles called Anyons exist with the properties at the end

  • depend on the space-time knot that their world lines have formed.

  • So you can kind of see now why I am so concerned

  • with whether the knots are the same or different.

  • We can conduct an experiment by which we create some particles,

  • move them around the form a knot

  • and then they have some property at the end of the knot.

  • Let me do the experiment again,

  • create the particles, make another knot, and I want to know

  • whether the properties of the particles at the end of the knot are the same

  • as the properties of the particles at the end of a different knot.

  • This is why I am concerned with whether the knots are the same or different.

  • Just looking at these two simple knots, it may not be obvious,

  • it sometimes is hard to tell

  • if two knots are the same or different.

  • Is there some way to unravel one, turn it into the other without cutting?

  • Fortunately mathematicians have been thinking

  • about this problem over 100 years, they cooked up some important tools

  • to help us distinguish knots from each other.

  • The most important tool is known as a Knot Invariant.

  • Knot Invariant is an algorithm that takes as an input

  • a picture of a knot and gives as an output

  • some mathematical quantity: a number, a polynomial,

  • some mathematical expressions, or mathematical symbols.

  • The important thing about a Knot Invariant is that equivalent knots,

  • 2 knots can be deformed into each other without cutting

  • have to give the same output.

  • If I have two knots - I don't know if they are the same or not -

  • I put them into the algorithm, and if they give two different outputs,

  • I know immediately that they can't be deformed with each other without cutting,

  • they are fundamentally different knots.

  • Now, in order to show you how these things work,

  • I'm actually going to show you how to calculate a Knot invariant.

  • The problem here is I have to give you a warning

  • that there's going to be math.

  • Now, I've given this talk at high schools before,

  • and nobody died.

  • (Laughter)

  • So, I suspect most people can handle this amount of math,

  • but some people are very math-phobic like the person in the slide.

  • If that's you, just close your eyes when you get scared,

  • open them up later, everything will be ok, you won't miss too much.

  • So, the Knot Invariant we are going to consider

  • is known as Kauffman Invariant or the Jones Invariant.

  • We start with a number which we call "A".

  • A stands for a number in this case.

  • The first rule of the Kauffman Invariant is if you ever have a loop of a string,

  • a simple loop with nothing go through it,

  • we can replace that loop with the algebraic combination

  • −A² − 1/A².

  • That combination occurs frequently, so we call it "d".

  • Anyway the first rule is then,

  • if you ever have a loop of string with nothing going through it,

  • you can replace that loop with just the number "d".

  • The second rule is a harder rule.

  • This rule says if you have two strings that cross over each other,

  • you can replace the picture

  • with the two strings crossing over each other

  • with the sum of two pictures.

  • In the first picture, the strings go vertically,

  • in the second picture, the strings go horizontally.

  • The first picture gets a coefficient of A on front,

  • the second picture gets a coefficient of 1/A on front.

  • This may look very puzzling

  • because you've replaced a picture with a sum of two pictures

  • and we've put numbers in front of those pictures.

  • Now we're talking about adding pictures together

  • as well as putting numbers in front of our pictures.

  • But all we're doing is we're doing math with pictures.

  • I'll show you it's not that hard by actually doing a calculation.

  • What we're going to do is we're going to take the rules,

  • and we're going to apply them to a very simple knot.

  • This very simple knot is a figure 8 looking thing.

  • Well, secretly we know it's actually just a loop of string,

  • and we folded it over to make it look like a figure 8.

  • But suppose we didn't know that, suppose we weren't so clever

  • to figure out that we could just unfold it and make it into a simple loop.

  • We would go ahead and try to calculate the Kauffman knot invariant

  • by following the algorithm.

  • So what you do is you look at the knot,

  • and you discover the two strings are crossing over each other

  • so I've circled that in that red box.

  • Now within that red box,

  • we have two strings crossing over each other

  • so we can apply the rule and replace those two strings

  • crossing over each other with a sum of two pictures.

  • In the first the two string go vertical, in the second two strings go horizontal.

  • In the first you have a coefficient of A on front,

  • in the second you have a coefficient of 1/A on front.

  • Now we just fill in the rest of the knot exactly like it is over on the left.

  • So now we replaced one picture

  • by a sum of two pictures with appropriate coefficients.

  • In these pictures there are still crossings in the knot down below

  • where I've now indicated them in blue, and we have to apply the Kauffman rule

  • to these crossings as well which we do exactly the same way.

  • Now we have a sum of 4 diagrams with appropriate coefficients.

  • At this point, we've gotten rid of all the crossings,

  • and we're left with only simple loops,

  • and simple loops by the first rule get a value of "d".

  • So each time we have a loop we replace it by a factor of "d".

  • So in the first picture, for example, there're two loops, so we get d²,

  • the second picture is just one big loop, a factor of "d" and so forth.

  • At this point, we are now down to only symbols, and no pictures are left,

  • so have "A"s and "d"s, so it's just some algebra at this point,

  • so you combine together some terms, then we use the definition

  • of "d" being −A² − 1/A² to replace this by −d, we get a d³ canceling a −d³

  • and at the end of the day we get "d". Yay!

  • (Laughter)

  • Why does this get Yay?

  • This is exciting for two reasons: first of all, it's exciting

  • because it's the end of the math, the second reason it's exciting,

  • it's because of the result giving us "d".

  • The reason it's interesting that we get "d"

  • is because at the beginning,

  • what we actually started with was just a simple loop.

  • We folded it over to make it look like a figure 8,

  • but it was a simple loop

  • and the Kauffman invariant of a simple loop is just "d".

  • Even though we folded it over to make it a lot more complicated

  • when we went through this algorithm at the end of the day we get "d".

  • That's how the knot invariants work.

  • We could've folded over a hundred times and made it look incredibly complicated

  • but still it would have given us "d".

  • So if we have these two knots here and we want to know

  • if they're the same or different, we put them into the algorithm,

  • and we get out two different algebraic results.

  • These results don't equal each other, and so we know immediately

  • these two knots are fundamentally different,