Related papers: Quiddity sequences for $\mathrm{SL}_3$-frieze patt…
Frieze patterns have an interesting combinatorial structure, which has proven very useful in the study of cluster algebras. We introduce $(k,n)$-frieze patterns, a natural generalisation of the classical notion. A generalisation of the…
The main goal of this paper is to prove several new results about frieze patterns and their equivalents, the quiddity (or $\eta$-)sequences and to obtain a formula giving the number of non-similar frieze patterns of given finite width.
Frieze patterns are numerical arrangements that satisfy a local arithmetic rule. These arrangements are actively studied in connection to the theory of cluster algebras. In the setting of cluster algebras, the notion of a frieze pattern can…
The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last…
Frieze patterns, as introduced by Coxeter in the 1970's, are closely related to cluster algebras without coefficients. A suitable generalization of frieze patterns, linked to cluster algebras with coefficients, has only briefly appeared in…
Friezes patterns are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their…
This note generalizes $\mathrm{SL}(k)$-friezes to configurations of numbers in which one of the boundary rows has been replaced by a ragged edge (described by a juggling function). We provide several equivalent definitions/characterizations…
Motivated by Conway and Coxeter's combinatorial results concerning frieze patterns, we sketch an introduction to the theory of cluster algebras and cluster categories for acyclic quivers. The goal is to show how these more abstract theories…
Frieze patterns are combinatorial objects that are deeply related to cluster theory. Determinants of frieze patterns arise from triangular regions of the frieze, and they have been considered in previous works by Broline-Crowe-Isaacs, and…
Our purpose with this paper is, in first place, to recast the space of quiddity sequences corresponding to usual frieze patterns as a different type of SET operad, and second to introduce and study $\mathfrak{M}$-quiddity sequences where…
In 2017, Michael Cuntz gave a definition of reducibility of quiddity cycles of frieze patterns: It is reducible if it can be written as a sum of two other quiddity cycles. We discuss the commutativity and associativity of this sum operator…
In this article, we construct SL$_k$-friezes using Pl\"ucker coordinates, making use of the cluster structure on the homogeneous coordinate ring of the Grassmannian of $k$-spaces in $n$-space via the Pl\"ucker embedding. When this cluster…
A frieze is an array of numbers obeying the unimodular rule. Coxeter showed that a frieze with integer entries corresponds to a triangulation. Recently, Holm and J{\o}rgenson introduced friezes of type $\Lambda_p$ which correspond to…
In this article we count tame $ SL_3 $- and $ SL_4 $-frieze patterns with width $ w $ over a finite field $ K $, as well as some tame $ SL_k $-frieze patterns for higher $ k $. Let $ n = w + k + 1 $. We consider the sets $ C_k(n) $ of…
Frieze patterns of numbers, introduced in the early 70's by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the…
A fundamental problem in spherical distance geometry aims to recover an $n$-tuple of points on a 2-sphere in $\mathbb{R}^3$, viewed up to oriented isometry, from $O(n)$ input measurements. We solve this problem using algorithms that employ…
We provide a characterization of infinite frieze patterns of positive integers via triangulations of an infinite strip in the plane. In the periodic case, these triangulations may be considered as triangulations of annuli. We also give a…
We define and study a continuous version of 2-frieze patterns, a combinatorial structure closely related with frieze patterns of Coxeter and Conway. We describe the relation of continuous 2-friezes with the moduli space of projective curves…
Frieze patterns (in the sense of Conway and Coxeter) are related to cluster algebras of type A and to signed continuant polynomials. In view of studying certain classes of cluster algebras with coefficients, we extend the concept of signed…
Tropical friezes are the tropical analogues of Coxeter-Conway frieze patterns. In this note, we study them using triangulated categories. A tropical frieze on a 2-Calabi-Yau triangulated category $\mathcal{C}$ is a function satisfying a…