Related papers: Ptolemy relations for punctured discs
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…
We give a complete classification of torsion pairs in the cluster categories associated to tubes of finite rank. The classification is in terms of combinatorial objects called Ptolemy diagrams which already appeared in our earlier work on…
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…
Motivated by the definition of super-Teichm\"uller spaces, and Penner-Zeitlin's recent extension of this definition to decorated super-Teichm\"uller space, as examples of super Riemann surfaces, we use the super Ptolemy relations to obtain…
In this article, we establish a relation between the values of a frieze of type D-tile and some values of an SL2-tiling t associated with a particular quiver of type A-tilde . This relation allows us to compute, independently of each other,…
This article, based on joint work with Gabriel Carroll, Andy Itsara, Ian Le, Gregg Musiker, Gregory Price, Dylan Thurston, and Rui Viana, presents a combinatorial model based on perfect matchings that explains the symmetries of the…
We present a combinatorial model for cluster algebras of type $D_n$ in terms of centrally symmetric pseudotriangulations of a regular $2n$-gon with a small disk in the centre. This model provides convenient and uniform interpretations for…
A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with…
Finite frieze patterns with entries in $\mathbb{Z}[\lambda_{p_1},\ldots,\lambda_{p_s}]$ where $\{p_1,\ldots,p_s\} \subseteq \mathbb{Z}_{\geq 3}$ and $\lambda_p = 2 \cos(\pi/p)$ were shown to have a connection to dissected polygons by Holm…
We give a geometric realization of cluster categories of type $D_n$ using a polygon with $n$ vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain…
Let Q be a quiver without loops and 2-cycles, let A(Q) be the corresponding cluster algebra and let x be a cluster. We introduce a new class of integer vectors which we call frieze vectors relative to x. These frieze vectors are defined as…
For each Dynkin diagram $D$, we define a ''cluster configuration space'' ${\mathcal{M}}_D$ and a partial compactification ${\widetilde {\mathcal{M}}}_D$. For $D = A_{n-3}$, we have ${\mathcal{M}}_{A_{n-3}} = {\mathcal{M}}_{0,n}$, the…
We introduce some new Frobenius subcategories of the module category of a preprojective algebra of Dynkin type, and we show that they have a cluster structure in the sense of Buan-Iyama-Reiten-Scott. These categorical cluster structures…
We investigate skein relations in cluster algebras from punctured surfaces, extending the work of \c{C}anak\c{c}i-Schiffler and Musiker-Williams on unpunctured surfaces. Using a combinatorial expansion formula by…
In this survey article we explain the intricate links between Conway-Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the…
In this article we consider tame $ SL_3 $-friezes that arise by specializing a cluster of Pl\"ucker variables in the coordinate ring of the Grassmannian $ \mathscr{G}(3,n) $ to $ 1 $. We show how to calculate arbitrary entries of such…
We continue our investigation on denominator conjecture of Fomin and Zelevinsky for cluster algebras via geometric models initialed in \cite{FG22}. In this paper, we confirm the denominator conjecture for cluster algebras of finite type.…
We study super cluster algebra structure arising in examples provided by super Pl\"{u}cker and super Ptolemy relations. We develop the super cluster structure of the super Grassmannians $\Gr_{2|0}(n|1)$ for arbitrary $n$, which was…
We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…