# Cluster Variables for non-convex n-gons

Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration of creating the new cluster variables using the Ptolemy's theorem, which also manifests other fundamental properties, such as positivity and Laurent phenomenon. (For example this http://arxiv.org/abs/1212.6263.)

Although I imagine one could address this question via the generalization of cluster algebras to other oriented surfaces with boundary, I was wondering if one could start with an arbitrary n-gon (not necessarily convex) and proceed from this point.

For instance, if one just deflates pentagon to an appropriate 5-gon, there are some diagonals without any flips and consequently the number of clusters we get would be different, so the cluster algebra varies and seems to be dependent of the deflation we apply to a fixed n-gon.

Is there any written work on this naive idea of generalization of cluster algebras from convex to arbitrary polygons, approachable for non-experts?

Clear answers and more importantly good references are highly appreciated.

• Perhaps you could provide a reference that shows "mutations of the diagonals of a convex n-gon" in this context? Thanks. – Joseph O'Rourke Jan 12 '16 at 23:35
• @JosephO'Rourke, thanks for your suggestion. I just added one which uses the aforementioned method after a few pages, to draw the associahedron. – Kaveh Jan 12 '16 at 23:51

Although I am unfamiliar with cluster algebras, there has been an attempt to generalize the associahedron, an object that records the structure of diagonalizations of a convex polygon, to "construct a polytopal complex analogous to the associahedron based on convex diagonalizations" of a non-convex polygon $P$:

Devadoss, Satyan L., Rahul Shah, Xuancheng Shao, and Ezra Winston. "Visibility graphs and deformations of associahedra." arXiv:0903.2848 (2009).

Here is one figure from the paper based on the non-convex octagon $P$ shown:

I see that there is another paper on roughly the same topic, but I haven't looked at it:

Braun, Benjamin, and Richard Ehrenborg. "The complex of non-crossing diagonals of a polygon." Journal of Combinatorial Theory, Series A 117.6 (2010): 642-649.

• Thanks for your answer and sharing the link. Although the problem addressed in this paper is an interesting generalization of Stasheff polytope, I assume it does not fully address my question about the cluster algebras. The ordering they put on $\pi(P)$ is with respect to the number of diagonals and how one convex diagonalization is obtained from the other by adding new diagonals, while for the cluster variables, some labeling of the diagonals of a triangulation of polygon is fixed and by the exchange formula (flip of a certain diagonal) the new cluster variable is obtained step by step. – Kaveh Jan 13 '16 at 1:57

Perhaps others will have suggestions, but I'm not aware of any cluster algebra papers that tackle this situation in the way you suggest.

One should be able to handle "diagonals without flips" as frozen variables (also called coefficients): one simply can't mutate there. Then, for the purposes of the cluster theory, all the action happens on the mutable bit, which I suspect simply corresponds to "trimming" the polygon to something convex. (This is a very rough idea in my head and might well be wrong...)

Something similar happening in nature can be seen in some work of mine with Sira Gratz (http://arxiv.org/abs/1212.3528), where we look at triangulations of an infinity-gon (in principle, a convex one). In that situation, you can get arcs that are not mutable and they then appear as coefficients, as I say.

• Thanks for sharing your idea and work. You're right, papers, as far as I know, begin with the formal definition- start with the initial seed and the exchange matrix and go through the iterative procedure of mutation to produce new variables.However, some lectures I have attended and lecture notes try to avoid the formal definition at the beginning. Your suggestion, considering more edges as frozen variables, makes perfect sense. However, behind the question I asked, I had a question in my head that I am not sure if I could phrase it properly. Nonetheless, I ask it in the following comment: – Kaveh Jan 13 '16 at 22:55
• "Staring from the regular configurations of a convex n-gon, we can continuously change this n-gon with exactly (n-3) non-frozen variables to any other configuration. As far as the result is convex, we have the same cluster algebra. But, once we get to concave configurations, the number of frozen variables changes, so we have different cluster algebras. My question is: Having a continuous change at the level of geometry, do we have any machinery to encode this continuous change at the level of cluster algebras generated for different configurations?" Sorry if the question is not well-phrased! – Kaveh Jan 13 '16 at 22:59
• I don't think so. The number of frozen variables is discrete, not continuous. I think the analogy would be that it just undergoes a "phase transition". – Jan Grabowski Jan 14 '16 at 10:51

In Fomin, Shapiro & Thurstons paper called Cluster Algebras and triangulated surfaces Part I: Cluster Complexes they consider the surfaces up to homeomorphisms. So wheter you consider a convex polygon or a deformed one does not really matter.

Furthermore if you leave one diagonal fixed, you could regard it as a boundary splitting the polygon into two smaller ones. The corresponding Cluster Algebra then would be the product (as in Fomin and Zelevinsky's initial paper Cluster Algebras: Foundation) of those two I guess.