Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

Sulanke, T.; Lutz, Frank H.

doi:10.1016/j.ejc.2008.12.016

Cited by 28 publications

(33 citation statements)

References 56 publications

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“…To triangulate a surface of genus 5 we also need at least 12 vertices. However, there are far more possibilities (751 593 as enumerated by Lutz and Sulanke [21]) than in the case of genus 6.…”

Section: Resultsmentioning

confidence: 99%

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Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

Schewe

2009

Discrete Comput Geom

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Abstract. We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R 3 . We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of polytopes.

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Section: Resultsmentioning

confidence: 99%

“…Additionally, we used our program to verify that all 821 minimal vertex triangulations of a surface of genus 4 as classified by Lutz and Sulanke [21] admit a chirotope.…”

Section: Methodsmentioning

confidence: 99%

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

Schewe

2009

Discrete Comput Geom

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“…Enumeration. When enumerating triangulations of manifolds of a given dimension with a fixed number of vertices or facets, we want to ensure that the objects produced are indeed manifolds and discard all others [6,7,31]. 2.…”

Section: Introductionmentioning

confidence: 99%

Heuristics for Sphere Recognition

Joswig

Lutz

Tsuruga

2014

Mathematical Software – ICMS 2014

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Abstract. The problem of determining whether a given (finite abstract) simplicial complex is homeomorphic to a sphere is undecidable. Still, the task naturally appears in a number of practical applications and can often be solved, even for huge instances, with the use of appropriate heuristics. We report on the current status of suitable techniques and their limitations. We also present implementations in polymake and relevant test examples.

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“…Such triangulations are tessellations of the convex hull of the point set into nonintersecting simplicial building blocks (triangles in two dimensions, tetrahedrons in three dimensions) [21]. Triangulations are an important tool in physics for describing curved space(-times), in quantum geometry (e.g., in the framework of causal dynamical triangulations [22] and in spin foams [23]); they are also a major object of study in topology and geometry where one is for example interested in the number of distinct triangulations of a given topological manifold [24]. Triangulations are commonly used to describe foams [25], where often a special so-called Delaunay triangulation, which is the dual to the Voronoi tessellation, is used for foam construction.…”

Section: Introductionmentioning

confidence: 99%

Unimodular lattice triangulations as small-world and scale-free random graphs

2015

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Real-world networks, e.g., the social relations or world-wide-web graphs, exhibit both small-world and scale-free behaviour. We interpret lattice triangulations as planar graphs by identifying triangulation vertices with graph nodes and one-dimensional simplices with edges. Since these triangulations are ergodic with respect to a certain Pachner flip, applying different Monte Carlo simulations enables us to calculate average properties of random triangulations, as well as canonical ensemble averages, using an energy functional that is approximately the variance of the degree distribution. All considered triangulations have clustering coefficients comparable with real-world graphs; for the canonical ensemble there are inverse temperatures with small shortest path length independent of system size. Tuning the inverse temperature to a quasi-critical value leads to an indication of scale-free behaviour for degrees ⩾ k 5. Using triangulations as a random graph model can improve the understanding of real-world networks, especially if the actual distance of the embedded nodes becomes important.

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Isomorphism-free lexicographic enumeration of triangulated surfaces and 3-manifolds

Cited by 28 publications

References 56 publications

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

Heuristics for Sphere Recognition

Unimodular lattice triangulations as small-world and scale-free random graphs

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