The Arithmetic of Braids (2022)

Summary

The Arithmetic of BraidsWhether in the context of hairstyles, friendship bracelets, or even parachute cords -- most will be familiar with the notion of a braid. As can be seen in the images above, each braid starts with some number of strands which are repeatedly crossed under/over each other in some way. Note that we typically don't allow the strands of a braid to "turn back up". We can represent the particular crossings of a braid with a braid diagram like the one shown below. Note the diagram shown describes a braid similar to (but longer than) the hair braid above. Of course, other braids will have a different number of strands and/or a different sequence of crossings. Some may even include sequences of crossings that don't even repeat, such as the one shown below: Taking inspiration from braids in the real world, "tugging" on the strands in one direction or another -- even when new crossings result (as long as we don't allow any one strand to pass through another) -- can lead to different representations of what is essentially the same braid. As an example, consider the following two diagrams which actually represent the same braid. While the two braid diagrams above represent the same braid, certainly the one on the left seems "simpler" in some capacity. This raises the question: "How does one simplify a given braid diagram?" Remember this question -- we'll come back to it in a bit. Admittedly, drawing braid diagrams like the ones previously seen -- and especially when they are not fully simplified -- can be tedious. However, there is a much easier way to represent braids! Towards this end, observe that if we "tug" in just the right places, we can always jiggle any particular crossing a little bit to the left or right, as desired. In this way, we can arrange any braid (with a finite number of crossings, anyways) so that no two crossings happen simultaneously as we scan the braid from left to right. As an example, consider the braid diagram involving 5 strands ...