Labeling the cells of a triangle-free chord diagram

Here's a simple way to label each of the regions of a triangle-free chord diagram by the indices of at most two of the chords.

Draw any triangle-free chord diagram in such a way that each chord partitions the disk in which the diagram is drawn into a region inside the chord and a region outside the chord, and such that one of the regions of the arrangement is outside all of the chords.

Now, consider the partial order in which chord \( x \) is a descendant of chord \( y \) if \( x \) lies entirely inside \( y.\) Below is a Hasse diagram for partial order formed in this way from the same example chord diagram.

The arrangement of the chords partitions the disk into regions; the set of chords containing any region is an upper set in the partial order. There must be either one or two minimal elements in this upper set (by the assumption that the diagram is triangle-free), and we can use these elements to label each region.

These labels have some very nice properties. Firstly, the labels consist exactly of the empty set, each chord, and each crossing pair of chords. Thus, the number of regions can be seen to be exactly the number of crossings, plus the number of chords, plus one. (This is true more generally without the assumption that the diagram is triangle-free, by Euler's formula, but not with as simple a bijective proof.)

Secondly, it is easy to determine the neighbors of any region. If the region is the empty set, its neighbors are labeled by the maximal elements in the partial order. If the region is labeled by a single element x, then its neighbors are (1) the region labeled by the immediate ancestors of \( x \) in the partial order, (2) the regions \( x,y \), where \( y \) is the smallest chord crossing \( x \) and the smallest chord with a label larger than \( x \) that crosses x, and (3) the regions \( y \) where \( x \) is the only immediate ancestor of \( y \) in the partial order. If the region is labeled by a pair \( x,y, \) then its neighbors are (1) the regions labeled \( x,z \) where \( z \) is adjacent to \( y \) in the sorted list of crossings of \( x \) and where \( y \) and \( z \) are both on the same side of \( x; \) (2) the regions labeled \( y,z \) satisfying a symmetric condition; (3) the region labeled \( x, \) if \( y \) is either the smallest chord crossing \( x \) or the smallest chord with a label larger than \( x \) that crosses \( x; \) (4) the region labeled \( y, \) if a symmetric condition is satisfied; and (5) the regions labeled \( z, \) where \( x \) and \( y \) are the two immediate ancestors of \( z \) in the partial order.

As a medium, the description of the regions and adjacencies between regions is even simpler. A region corresponds to an upper set in the partial order that is either empty, generated by a single chord, or generated by a crossing pair. Two regions are adjacent if the corresponding upper sets differ by the addition or removal of a single element.

P.S. I realized after posting this that I forgot to draw chord 12. I'll add it if I ever re-use these figures for a paper, but for now, just pretend it's there.