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,