Yesterday I wrote about the maypole braid group and left two questions for homework. Here are the solutions.
The first question was to show that the following maypole braids could not be represented as products of .
Observe that each time we apply , one strand moves clockwise and one moves counterclockwise. Thus there is no net rotation among all the ribbons. More concretely, we could assign to a ribbon for each counterclockwise rotation and for each clockwise rotation. If we sum those quantities over the entire braid we must obtain zero. However, since the first example above has a net rotation of the second one , and the third one they cannot be obtained in this manner.
Of course each time we apply this adds one to the net rotation. I showed yesterday that we can write the first two maypole braids using the and . That brings us to the second homework question which was to write the third maypole braid as such a product. The answer is: .
I’m hoping to write another follow-up post about this topic. I’ve spent the last few days scouring the internet and MathSciNet to see what is known about these groups. Not surprisingly, I was not the first to discover them (although I’m probably the first to apply them to maypoles!). In the literature they are known as annular braids, and they have connections to Artin groups and Coxeter groups. Hopefully I’ll have the time to read, digest and write about what I’ve learned.