The dance around the maypole is an ancient tradition probably originated in Germanic cultures
Nowadays it is performed in folks festivals or as a children's game in many European and American countries
The dancers twist the colored ribbons following precise movements
Then they perform an inverse dance to undo the braid
The ribbons can be braided in different ways and new choreography can generate new braids
In fact dances and braids have a lot to do with each other
Each of these dancers arrives in a colored position after each piece of the dance
So, at the end, they occupy the same positions as at the beginning of the dance
How can we keep track of the movements and make a static drawing that describes the dance?
Let's formalize the dance
Turn the dancers into points
Let the points move in a disc
At each moment the points are distinct: the dancers never collide!
The points go back to their initial positions, but possibly they exchange places
If we draw the path described by each point on the disc we can get a very complicated drawing
We see some intersections and we don't know who passed there first So we can not reconstruct the dance
A solution is to move the disc while the dance is being performed
In other words, we transform time into a spatial dimension, in this case the vertical direction
Of course in each moment we have the same number of dancers
This means that at each level a horizontal disc will meet each path exactly once
We have already seen diagrams like this: this is a braid!
Following our convention, we should align the braid horizontally
But just for this chapter we will change notation and draw braids vertically
Vice versa, given any braid we can turn it into a dance
In other words, we now have a description of braids as the trace of dancing points
Braids and dances are the same thing!
Now let's consider some particular dances where dancers are in pairs, always holding hands
As before, all the positions held in the beginning have to be filled at the end of the dance
This time, a pair of dancers becomes an arc
The arcs move keeping the endpoints on a disc
They can turn,
exchange positions,
pass one over other, and so on.
As before, we can translate the dance into a braid
We move the disc vertically and draw the paths described by the dancers, the endpoints of the arcs
Which braids are dances of couples?
Of course, the non-dance, where all the points stay still, is of this type So, the identity braid is a dance of couples
Performing a dance after another corresponds to composition of braids
Composing two dances of couples we get a new dance of the same type
Also the inverse of a dance of couple is of the same type
Then we say that braids describing dances of couples form a subgroup of the braid group. It is called the Hilden subgroup
There are some simple dances in the subgroup:
The two dancers within a single couple can exchange positions
Two couples can exchange positions
A couple can pass under the arms of another couple
Many other dances can be performed, but in fact the three movements just described are sufficient to assemble all Hilden's dances
In other words, these are the generators of the Hilden subgroup
Finding the relations, as in the case of the braid group, is much more complicated
What is all this good for?
Recall that in chapter 3 we described a way to relate braids to knots, via the closure
When we close a braid, we are actually adding new trivial strands and connecting each old strand with a new one, obtaining a knot
If we move the new strands behind the old ones we obtain a different braid, closed in a new way
This closure can be done on each braid with an even number of strands It is called the plat closure
Again, closing a braid as a plat we obtain a knot
Recall the Alexander theorem: any knot can be obtained by closing a braid
We saw how to cut a knot to obtain a braid
From the closed braid we can obtain a plat, thus any knot can be obtain as a closure of a plat
Can we write a Markov theorem for the new closure? That is, which braids give the same knot via plat closure?
We can do some simple moves:
we can compose our plat with Hilden elements both on the bottom
or on the top
Moreover, we can add two strands and a crossing
And go on like this, using Hilden elements and the new stabilization move
When we close the plat, the knot type will not change: we can retract the arcs and obtain the initial knot
Birman showed that these moves are enough:
two plats give the same knot if and only if they can be turned into each other through a sequence of moves of the two types
So the Hilden subgroup, that is the dances of couples, is one of the main ingredients of this theorem!
Now a question arises: how can we recognize the Hilden elements?
For example, this is a dance of couples
This is a Hilden braid, too
This is not, since single dancers of different couples can not exchange positions
But in general, given a braid with an even number of strands how can we check if it belongs to the Hilden subgroup?
This is one of the classic problems in combinatorial group theory
In the case of the Hilden subgroup there is a way to answer That is, there is an algorithm
But in general, in less peculiar groups, there can not exist such an algorithm!
Again, we have bumped into an algorithmic problem!
And again, we have found a connection with knot theory