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