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