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