Cluster structures on triangulated non-orientable surfaces

In 2002, Fomin and Zelevinsky introduced a cluster algebra; a dynamical system that has already proved to be ubiquitous within mathematics. In particular, it was shown by Fomin, Shapiro and Thurston that some cluster algebras arise from orientable surfaces. Subsequently, Dupont and Palesi extended this construction to non-orientable surfaces, giving birth to quasi-cluster algebras. The finite type cluster algebras possess the remarkable property of their exchanges graphs being polytopal. We discover that the finite type quasi-cluster algebras enjoy a closely related property, namely, their exchange graphs are spherical. Revealing yet more connections we unify these two frameworks via Lam and Pylyavskyy's Laurent phenomenon algebras, showing that both orientable and non-orientable marked surfaces have an associated LP-algebra. The integration of these structures is attempted in two ways. Firstly we show that the quasi-cluster algebras of unpunctured surfaces have LP structures. Secondly, to obtain a connection for all marked surfaces, we consider laminations, forging the notion of the laminated quasi-cluster algebra. We show that each marked surface exhibits a lamination which supplies the laminated quasi-cluster algebra with an LP structure

Number of Triangulations of a M\"obius Strip

Consider a M\"obius strip with $n$ chosen points on its edge. A triangulation is a maximal collection of arcs among these points and cuts the strip into triangles. In this paper, we proved the number of all triangulations that one can obtain from a M\"obius strip with $n$ chosen points on its edge is given by $4^{n-1}+\binom{2n-2}{n-1}$, then we made the connection with the number of clusters in the quasi-cluster algebra arising from the M\"obius strip

Shellability and Sphericity of Finite Quasi-arc Complexes

Quasi-triangulations of a non-orientable surface were introduced by Dupont and Palesi (J Algebr Comb 42(2):429–472, 2015). The quasi-arc complex provides an intricate description of the combinatorics of these quasi-triangulations. This is the simplicial complex where vertices correspond to quasi-arcs and maximal simplices to quasi-triangulations. We prove that when the quasi-arc complex is finite then it is shellable and, as a consequence, it is homeomorphic to a sphere