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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 decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics., 9 pages, 6 figures

We give a geometric realization of the module category of a hereditary algebra of type A~A~. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type A~A~. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.

In this article, gentle algebras are realised as tiling algebras, which are associated to partial triangulations of unpunctured surfaces with marked points on the boundary. This notion of tiling algebras generalise the notion of Jacobian algebras of triangulations of surfaces and the notion of surface algebras. We use this description to give a geometric model of the module category of any gentle algebra., Comment: 21 pages, 23 figures

We derive a formula for the number of flip-equivalence classes of tilings of an n-gon by collections of tiles of shape dictated by an integer partition λ. The proof uses the Euler–Poincaré formula; and the formula itself generalises the Euler–Poincaré formula for associahedra

We derive a simple bijection between geometric plane perfect matchings on $2n$ points in convex position and triangulations on $n+2$ points in convex position. We then extend this bijection to monochromatic plane perfect matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of convex point sets. These structures are related to a generalization of Temperley-Lieb algebras and our bijections provide explicit one-to-one relations between matchings and tilings. Moreover, for a given element of one class, the corresponding element of the other class can be computed in linear time., Comment: 12 pages, 8 figures

We associate strand diagrams to tilings of surfaces with marked points, generalising Scott's method for triangulations of polygons. We thus obtain a map from tilings of surfaces to permutations of the marked points on boundary components, the {\em Scott map}. In the disk case (polygon tilings) we prove that the fibres of the Scott map are the flip equivalence classes. The result allows us to consider the size of the image as a generalisation of a classical combinatorial problem, and hence to determine the size in low ranks., Comment: (v2): Section 8 added where we show bijections between tilings, rhombic plabic graphs and minimalist strand diagrams. (v3): Added appendix (with Max Glick) computing the asymptotic growth rate of the image of the "Scott map"

We give a geometric model for a tube category in terms of homotopy classes of oriented arcs in an annulus with marked points on its boundary. In particular, we interpret the dimensions of extension groups of degree 1 between indecomposable objects in terms of negative geometric intersection numbers between corresponding arcs, giving a geometric interpretation of the description of an extension group in the cluster category of a tube as a symmetrized version of the extension group in the tube. We show that a similar result holds for finite dimensional representations of the linearly oriented quiver of type A-double-infinity., Comment: 15 pages, 7 figures. Discussion of maximal rigid objects and triangulations at end of Section 3. Minor corrections