Frank H. Lutz
We discuss different approaches for the enumeration of triangulated surfaces. In particular, we enumerate all triangulated surfaces with 9 and 10 vertices. We also show how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly.
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We present a fast enumeration algorithm for combinatorial 2- and 3-manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3-manifolds with 11 vertices. We further determine all equivelar polyhedral maps on the non-orientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the non-orientable surfaces of genus 5 and 6.
We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for the case where the number of vertices is at most 11 and we get theoretical restrictions on genus-surfaces allowing us to get the list of triangulations of closed surfaces with at most 11 vertices.
We describe an algorithm for the enumeration of (candidates of) vertex-transitive combinatorial $d$-manifolds. With an implementation of our algorithm, we determine, up to combinatorial equivalence, all combinatorial manifolds with a vertex-transitive automorphism group on $n\leq 13$ vertices. With the exception of actions of groups of small order, the enumeration is extended to 14 and 15 vertices.
We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type $A_n$ or $A_{\infty}$ is given via rotation of Ptolemy diagrams.
We give explicit realizations with small integer coordinates for all triangulated tori with up to 12 vertices. In particular, we provide coordinate-minimal realizations in general position for all triangulations of the torus with 7, 8, 9, and 10 vertices. For the unique 7-vertex triangulation of the torus we show that all corresponding 72 oriented matroids are realizable in the 6x6x6-cube. Moreover, we present polyhedral tori with 8 vertices in the 2x2x2-cube, general position realizations of triangulated tori with 8 vertices in the 2x2x3-cuboid as well as polyhedral tori with 9 and 10 vertice…
We prove that every triangulation of either of the torus, projective plane and Klein bottle, contains a vertex-spanning planar Laman graph as a subcomplex. Invoking a result of Kir{á}ly, we conclude that every $1$-skeleton of a triangulation of a surface of nonnegative Euler characteristic has a rigid realization in the plane using at most 26 locations for the vertices.
We construct geometric realization for non-exceptional mutation-finite cluster algebras by extending the theory of Fomin and Thurston to skew-symmetrizable case. Cluster variables for these algebras are renormalized lambda lengths on certain hyperbolic orbifolds. We also compute growth rate of these cluster algebras, provide positivity of Laurent expansions of cluster variables, and prove sign-coherence of c-vectors.
The search for universality in random triangulations of manifolds, like those featuring in (Euclidean) Dynamical Triangulations, is central to the random geometry approach to quantum gravity. In case of the 3-sphere, or any other manifold of dimension greater than two for that matter, the pursuit is held back by serious challenges, including the wide open problem of enumerating triangulations. In an attempt to bypass the toughest challenges we identify a restricted family of triangulations, of which the enumeration appears less daunting. In a nutshell, the family consists of triangulated 3-sph…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric side, this requires opening the surface at each interior marked point into an additional geodesic boundary component. On the algebraic side, it relies on the notion of a non-normalized cluster algebra and the machinery of tropical lambda lengths. Our model allows for an arbitrary choice of coefficients which translates into a choice of a family of integral lami…
We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that the generating function of closed walks maps to the grand partition function of a system of particles with exclusion statistics of order $g=2$ and an appropriate spectrum, along the lines of a connection previously established by two of the authors. Reinterpreting the results in terms of the standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) a…
We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space. We show there is a simplicial 3-ball with 8 vertices that is collapsible, but also collapses onto the dunce hat, which is not collapsible. This 3-ball is as small as possible, because all triangulated 3-balls with fewer vertices are extendably collapsible. As we will see, the Alexander dual of the dunce hat is collapsible.
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