Dancing braids
2023--2024 -- my MSc project on using braids to model
Scottish country dancing figures. In a nutshell: imagine parking
a Maypole at the centre of the dance set and clipping a Maypole
ribbon to the waist of each dancer (in such a way that the
ribbon grows longer as necessary...): what is the pattern
created on the Maypole, and how can we use MATHS to analyse
it?
Introduction
Building foundations
Final dissertation
Q&A session / discussion
- Q: Can you classify "transparent" braid invariants? We
can think of braid invariants as being more about how the
braid acts on the space around it (stirrers in a viscous fluid
/ braid as a topological space), or
more about how the braid strands interact with each other (or neither). The
latter type (strand interactions) are most relevant (or transparent) for the dance
application.
- A note on Artin combing and the isomorphismin this way
In Chapter 3, we need to show that if we designate one strand of a braid to be the "Maypole", we can guarantee to be able to pull this strand straight whenever it starts and finishes in the same position. We present Artin combing as an algorithm to do this, and refer to Artin's 1947 paper for proof that Artin combing will work. In fact, it is possible to jump straight from the algebra used in the paper to an algebraic proof of the result.
- A note on the Brunnian chains In Chapter 6.1, we
claim that neither the n-Bonio nor the n-Horseshoe can be
danced with n dancers. This can be shown by finding a
general formula for the HOMFLY polynomial of the links and
applying
the Morton-Franks-Williams
inequality. We have written up these results in draft format:
more here
- Braidlab
When I started the project, and was "playing" with braids, I
used the braidlab Matlab library to explore
some properties of the braids.
-
Braidlab and annular braids: braidlab
partially supports annular braids: braidlab's annular braids
can have the n/1 crossing. However, there is no support for
the move I have called "twist", which is a definite limitation
-- in the project, we show that it is possible to model the
extra crossing in terms of twists, but not vice
versa. Furthermore, braidlab's implementation of the annular
braid (including depiction) is very much as an isomorphic
Artin, and the invariants available are all implemented
accordingly -- in particular, the "Maypole" closure is used
for properties of the closure, such as the Alexander
polynomial. However, as discussed in the project, this is not
necessarily the best option.
-
Braidlab and jiggles to meet the geometric
braid condition, dancers (or strands) must always be in
one of the "home positions" after a crossing. Since they
move in a continuous fashion, they will of course not be
in the home positions as they move through the crossing.
The practical implementation of the projection from a geometric
braid onto an Artin braid projects the strands onto a line, and then
forces the "snap to grid", pushing the strands to the nearest home
positions along the Artin braid. This has the effect that if
two strands are almost coincident at the projection angle, they may be
perceived as having crossed when they jiggle very slightly, as in the
example in the final presentation (indeed, the braids in the example were
generated with braidlab). As noted in the report, if the braid
condition were established on the geometric braid before projection,
no crossing would be perceived for this kind of jiggling -- but,
particularly due to the continuity, it would be very tricky
to implement this in practice.
Additions and extras
