This is a demonstration of the connections between triangulated polygons, paths in the Farey graph, Conway–Coxeter friezes, and cluster variables. For the background infromation on this, see our research and references therein.
The picture shows a polygon dissected into triangles, i.e. a triangulation. There are two ways to modify the triangulation: the first is by left-clicking on a diagonal to flip it; the second is using the itinerary. Right-click to make the diagonal negative; this affects the frieze, see below.
The itinerary of the corresponding path in the Farey graph, i.e. the counts of the triangles at each vertex, is shown to the right of the picture. Click on a 1 button to insert a 1 into the itinerary, which corresponds to attaching a triangle to a side of the triangulation, or click on a x button to remove a 1 from the itinerary, which corresponds to removing an external triangle from the triangulation. One possible choice of the vertices of the path in the Farey graph is shown below the corresponding itinerary entries.
That path in the Farey graph together with the edges in its interior is shown as the sequence of hyperbolic lines in the upper half-plane model of the hyperbolic plane. Superimposed are the corresponding Ford circles. When the path has even length, the circles' radii can be alternatingly multiplied by a positive factor preserving their tangencies; move the slider to see this.
There is a tame integer frieze corresponding to the triangulation with prescribed signs of diagonals. The fundamental domain of that frieze under the action of its glide reflection symmetry is shown. Every entry in the frieze corresponds to a pair of vertices in the triangulation, hover or tap to highlight the corresponding diagonal. The entries corresponding to the diagonals in the triangulation (all equal to 1 for positive diagonals or −1 for negative diagonals) have shaded background; these entries also correspond to the initial cluster variables, see below. The quiddity of the frieze (i.e. the row right below the row of 1s) is equal to the itinerary of the triangulation if all diagonals are positive. An integer frieze with negative entries still corresponds to some path in the Farey graph. That path has itinerary and vertices different from the ones shown, however they still can be read out directly from the frieze itself.
Every pair of vertices in the triangulated n-gon also corresponds to a cluster variable in the cluster algebra of type An−3. The triangulation defines a particular choice of initial cluster variables. All cluster variables are shown in the bottom part of the page. Again, the corresponding diagonal in the picture is highlighted upon hover or tap, and the list of the initial cluster variables xi is additionally given for convenience. The entries of the frieze are obtained by evaluating the corresponding cluster variables at xi = 1 or −1, so the entire frieze can be restored from these values only.
Cluster variables are known to exhibit the Laurent phenomenon, which in particular means that evaluating them at xi = 1 or −1 always results in integer numbers. However when some initial cluster variables are set to be equal to −1, which can be done in this app by making diagonals negative, the Schmaurent phenomenon arises: some of the remaining initial cluster variables are no longer present in the denominators of the Laurent polynomials. Thus, they can be set to be equal to any integer number, still giving integer results upon the evaluation of cluster variables. Although this app can only set such initial cluster variables to 1, the entries of the frieze corresponding to them are boxed to indicate that these values can be replaced by any integer, and it would still be possible to restore the entire tame integer frieze from these entries.