ScalingStacks

Example 3.2.5. The elements of $\tilde{L}(\sigma)$ are in bijection with intersection points between strands of a “good diagram” representing $\sigma$. Here, we define a strand diagram to be good if no more than two strands intersect at a given point and if the diagram minimizes the total number of intersection points. Similarly, the elements of $L(\sigma)$ correspond to intersections in an unfolded good strand diagram.

These descriptions can be deduced from Lemma 6.2.3 below, that shows those statements hold for pairs of strands. Now, the intersection point set for a good diagram is the disjoint union over intersection sets between pairs of strands, and a good diagram minimizes the intersection number among good diagrams if and only of each pair of strands minimizes its intersection number.

For example: