This movie is about braids and mathematics
Everyone knows what a braid is
Braids are everywhere:
in hairdressing
in jewellery
in leather belts, in ropes
in cakes, in cheese,
in bread
and in many other objects
Braiding is one of the oldest ways to decorate objects
And in mathematics? What is a braid?
Why do mathematicians study braids? And how?
Take a disc and choose some points inside it
Make a parallel copy of it
and start drawing strands that connect the marked points
The strands flow from the left to the right
They are not allowed to turn back
The strands can be braided and linked together
but two strands can not pass in the same point: intersections are not allowed
Draw strands until all the marked points are connected by them
This is a braid
For simplicity we will not draw the discs and the points anymore, but only the strands
In this way we can construct a lot of braids
just using two strands and repeating the same crossing a different number of times
With three strands we can realize more complex braids
And so on, with any number of strands
How can we recognize when two braids are combed in the same way?
The only thing that matters is the way in which the strands are linked together
For example, these two braids are the same:
we can deform one into the other, keeping the endpoints fixed and not letting the strands cross
So, we can represent one braid in many different ways
All these braids are obtained deforming the original one keeping the endpoints fixed
All these are equivalent braids.
Still, there are braids that can not be deformed one into the other
So we say that there are different equivalence classes formed by the braids that can be transformed into each other
To determine a class we can choose any of its representatives
When we draw a braid, we do not mean just that specific representative but all the braids it represents
How can we choose the representatives for each braid?
Further, if we draw two braids, how can we know if they are equal or different?
For example, these two braids are the same: we just need to move the blue strand between the red one and the green one
But what about these two? Are they the same braid?
And if we have more complex braids?
To answer, we try to find a mathematical structure on the set of braids
The first thing we note is that we can compose two braids
This just means to put one after the other and connect the strands
Now, take two representatives of the same braid
On the right draw two representatives of another braid
Then compose them
Making some deformations we see that these are two representatives of the same braid
The result does not depend on which representatives we choose for our braids
The composition of braids is an operation, like the product of positive numbers
The product has some beautiful properties:
it is associative
and commutative
Do these properties hold for the composition of braids?
Here we compose three braids in the two possible ways keeping the order
Of course we get the same braid!
So, associativity holds
and we can write any number of subsequent compositions without using parentheses
What about commutativity?
Here is an example: we compose two braids in the two possible ways
The results can be different!
Look at the red strand, starting from bottom left: it arrives in different positions on the right
So these braids can not be the same. This means that commutativity does not hold.
Thinking of the product again, there is a neutral element, 1
Is there such an element for the composition of braids?
Take the trivial braid, where all the strands are parallel
When we compose any braid with it, the result is equivalent to the original braid
Any number has an inverse with respect to the product
Is there an inverse for a braid?
Look at the braid in a mirror
We can compose these two braids in two ways: first the original braid and then its mirror image or vice versa, first the mirror image and then the original braid
In both cases we can simplify them and we get the identity braid
So, for every braid there is an inverse
So far we have seen that on the set of braids we can define an associative operation
This has a neutral element
and for every element in the set there is an inverse
Such a structure is called a group
To deal with braids it can be useful to associate a word, a sequence of symbols, to each braid
To the identity braid we assign the symbol 1 because this is the neutral element for the composition
We call sigma_1 the braid where the first two strands are exchanged by a clockwise twist looking from the left
The other strands are straight
In sigma_2 the second and the third strands are linked together by a clockwise twist
And here is sigma_3 where the linked strands are the third and the fourth
If the first two strands twist in the other direction, we will have sigma_1 inverse
It is called so because it is actually the inverse of sigma_1
This is sigma_2 inverse, the inverse of sigma_2
And sigma_3 inverse, the inverse of sigma_3
Using these elementary braids as bricks we can construct many different braids
Here is an example: we put a brick after the other and write the corresponding word
The braid is described by the word
Can we describe any braid using the elementary braids?
Take this braid. It looks quite complicated
But we can deform it
Then we can cut it into levels
Each level is an elementary braid, with just one crossing
This procedure works on every braid, so to any braid we can associate a word
Now, there are different words representing the same braid
For example, an elementary braid and its inverse cancel out when they are side by side
We can replace them with the identity braid
And we can cancel out the trivial piece
Vice versa, we can insert in any place an elementary braid followed by its inverse
And there is more: here the blue strand passes between two strands that form a crossing
Here two crossings involving different strands exchange their positions moving horizontally
All these movements are easily translated into manipulations of the words
Surely there are other movements:
here the yellow strand passes in front of a crossing
Here we cancel out an elementary braid and its inverse even if they are distant
How can we find all the possible manipulations of the words? It seems an endless job
Don't worry, mathematicians have already solved this problem
The first mathematician who studied braids in this terms was Emil Artin
In 1928 he wrote the first paper about braids, in German
Twenty years later he wrote a second important one in English
Emil Artin was the first who described the structure of group on the set of braids and noted that the group can be described in this way
This mathematical notation means that any braid can be described as a word in the sigma_i's and their inverses
In every group we have these reductions. They are local: they change just a little part of the word, keeping the rest fixed
The only other substitutions permitted in a word are of two kinds. We already saw the corresponding moves:
moving the blue strand through a crossing
and exchanging the left and the right crossing
With these two moves, called relations, we can generate all the words that describe the same braid
We have turned braids into mathematical objects!