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Let's change topic for a while
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These sailor knots are not knotted!
The knots can escape from the string ends
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To capture the knots we have
to glue the string ends together
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Now these are knots
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Closing the ends of the objects on the right we get more complex knots, where two components can be linked together
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Without originality, we call these objects links
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If we can untie them, we obtain a very simple link, called unlink
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From now on, we will say knots referring to both knots and links
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Look at this knot.
If we take its mirror image, it looks different
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but we can deform the knot on the left into the one on the right
without cutting the string
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Then, we will consider them the same knot
even if they look different at first
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These two knots look different, too.
But they are tied in the same manner
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They have different shapes but represent the same knot
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One can be deformed into the other
without ever cutting the strings
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This knot is called trefoil
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As before, we can ask: is the trefoil the same as its mirror image?
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This is a very technical and difficult problem
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The two knots are not the same.
They are called left-hand and right-hand trefoil
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How can we distinguish different knots
or recognize different drawings of the same knot?
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A way to approach such a problem is to relate
the realm of knots, closed strings, to that of braids
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It is easy to see that when we have a braid,
we can tie the strand ends together
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and we can cross the bridge to the realm of knots
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And vice versa?
Can we cross the bridge in the other direction?
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A theorem of Alexander ensures that it is possible
and gives an algorithm to do it
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We describe it, even if more efficient ones are known
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We choose an axis.
We will make a reel around it
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We choose a starting point
and walk along the knot turning clockwise around the axis
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At some time the knot can turn and we will be walking anti-clockwise
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We color all the anti-clockwise pieces red
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Now we move each red piece in turn to the other side of the axis
and color them yellow again
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In the end we have turned our knot into a reel around the axis
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Walking along the knot
we will always be going in the same sense around the axis
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We take a half plane with the axis as border
and cut the knot along it
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We open the strands
keeping the endpoints fixed on the half planes
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The strings can never touch each other
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And here is our braid!
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When we close it, we get a knot equivalent to the original one
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that is, we can deform one into the other without cutting the string
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Why make life so difficult? The knot on the left seems simpler!
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But in this way we can exploit
the group structure that we know on braids
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Alexander's theorem ensures that
we can obtain any knot as the closure of a braid
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but two braids can be very different and still give the same knot
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For example, they don't even need
to have the same number of strands!
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So the question now is: given two arbitrary braids,
do they give the same knot, once they are closed?
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We introduce a new operation, called conjugation
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Choose a braid
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Take another one and its inverse
and compose them in this manner:
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one on the left and the inverse on the right
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The new braid is called a conjugate of the first
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Note that the corresponding operation with numbers will not change the starting number: the product is commutative
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On the contrary, two braids can be different and still be conjugated
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Here is a simpler example: two generators of the braid group
are surely different braids
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But look: they are conjugated
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In general understand whether
two given braids are conjugate is an intriguing issue
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Let's go back to our problem:
when do two braids close to the same knot?
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If we conjugate a braid with any other,
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when we close the new braid
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we can shift the lateral pieces so that they cancel out
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since one is the inverse of the other
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In this way we get the same knot as closing the original braid
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We can modify our braid in another way:
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add a strand on the top
and link the two top strands together
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The new braid, when closed,
will give the same knot as the old one:
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we just need to undo the loop
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Of course we can do vice versa, too:
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cancel the last strand
if it links just once with the second-to-last
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These operations are called stabilizations
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A Russian mathematician, Markov, noticed that:
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Two braids give the same knot if and only if they are related by a sequence of moves of the two kinds we have just seen
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This is now known as Markov theorem, even if the first proof is probably due to one of his students
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We didn't show the difficult part of this theorem,
namely, that the two kinds of moves are enough
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We just make an example. We already know that
these two braids give the same knot
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Now we can prove this, without passing through the realm of knots!
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We have to find a sequence of conjugations and stabilizations
that transforms one braid into the other
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Markov theorem exactly says when two braids give the same knot,
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but in this form it is of no concrete use:
finding a sequence of relations can be very difficult
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And, as in chapter 2, if we can't find such a sequence,
it doesn't mean that none exists!
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Approaching knots through braids seems not to simplify things
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But one of the major results on knots
was achieved just thanks to the braids!
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In 1984 Jones, studying braids, proved an outstanding result
that revolutionized the theory of knots!
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It was so important that he won the Fields medal for it,
the most important award for mathematicians
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Jones found a way to associate a formula, a mathematical expression, to each braid
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The powerful fact is that this permits us to distinguish
the knots obtained closing the braids:
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if two braids have different formulas, then they give different knots
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On the other hand, if two braids differ by Markov moves,
then they are associated to the same formula
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This means that the formula, called Jones polynomial,
only depends on the knot and not on the braid used to get it!
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As an example, the trefoil and the figure eight knot have different Jones polynomials, so they are surely different knots
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Later another algorithm to calculate the Jones polynomial
was discovered, not involving braids anymore
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Choose a direction to walk along the knot
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There are places where we see a crossing
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The crossings can be of two types
depending on the strand that passes behind the other
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Resolving a crossing means to break the arcs
and connect them in the other way, respecting their orientation
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Introduce a relation between these pieces
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The symbol V indicates the Jones polynomial
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Now, associate the polynomial 1 to the unknot
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Using just these two relations,
we can calculate the polynomial on every knot
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Choose a crossing and apply the first relation to it
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Simplify...
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and apply the first relation again
to the knot on the right to write a new equation
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Choose a new crossing and go on like this,
writing equations and simplifying
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Using always the same relations, we can calculate
the Jones' polynomial of the simplest knots
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Then, going back step by step, we can reconstruct
the expression for the complex knots
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in our case the right trefoil
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We didn't check that this machinery is coherent, that is, making different choices always gives the same expression for a fixed knot
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This is the difficult part, and the interesting one:
the Jones' polynomial is an invariant of knots:
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calculated on two equivalent knots, it is the same
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If we calculate the Jones polynomial on the left trefoil, we obtain
an expression that is symmetric to the other, in some sense
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But not equal
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We have proved that the two trefoils are not equivalent!