Strand diagrams

 Take a cylinder with a set of “marked points” on its upper and lower boundary segments, both of which are labelled from 1 to \(n\). We get a strand diagram by drawing some “strands” on the cylinder. Each strand should start at a lower marked point and end at an upper marked point, or start at an upper marked point and end at a lower marked point. There are a few extra conditions that the strands need to satisfy in order to be a strand diagram, and there are even more conditions that a strand diagram can satisfy in order to be “nice.” We often represent these diagrams by drawing them in a rectangle, and identifying the left and right sides to make a cylinder.

When we look at a strand diagram, we care about how far each strand moves to the left or right from its start to finish. We can encode this information in a pair of affine permutations. We would like to classify the pairs of affine permutations which we can get from strand diagrams, and try to relate strand diagrams which have the same pair of affine permutations. There are many “elementary” strand diagrams which have few intersections. By stacking these simple strand diagrams on top of each other, we can obtain more complicated strand diagrams with more complicated pairs of affine permutations. We will use this strategy to begin answering these questions.