Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userQuiddities of polygon dissections and the Conway-Coxeter frieze equation
97
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study a $2 times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $mathrm{PSL}(2,mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
rate research
Read More
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,mathbb{Z})$.
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
Let $C_{k_1}, ldots, C_{k_n}$ be cycles with $k_igeq 2$ vertices ($1le ile n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the M{o}bius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, but also give an explicit relation matrix among these edges. So we obtain a uniform method to compute the sandpile group of arbitrary (twisted) polygon rings, as well as the number of spanning trees of (twisted) polygon rings. As an application, we compute the sandpile groups of several infinite families of polygon rings, including some that have been done before by ad hoc methods, such as, generalized wheel graphs, ladders and M{o}bius ladders.
We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type ${rm C}_{2}$ and ${rm A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105-1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and SL-friezes.
We explain the notion of $q$-deformed real numbers introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
