Frank H. Lutz

Simplicial manifolds, bistellar flips and a 16-vertex triangulation of the Poincaré homology 3-sphere

We present a computer program based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16-vertex triangulation of the Poincaré homology 3-sphere; we construct an infinite series of non-PL. d-dimensional spheres with d + 13 vertices for d ≥ 5; and we show that if a d-manifold, with d ≥ 5, admits any triangulation on n vertices, it admits a noncombinatorial triangu lation on n + 12 vertices.

Random Discrete Morse Theory and a New Library of Triangulations

We introduce random discrete Morse theory as a computational scheme to measure the complexity of a triangulation. The idea is to try to quantify the frequency of discrete Morse matchings with few critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its naiveté, this approach turns out to be very successful even in the case of huge inputs. In our view, the existing libraries of examples in computational topology are “too easy” for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.

A CENSUS OF TIGHT TRIANGULATIONS

A triangulation of a manifold (or pseudomanifold) is called a tight triangulation if any simplexwise linear embedding into any Euclidean space is tight. Tightness of an embedding means that the inclusion of any sublevel selected by a linear functional is injective in homology and, therefore, topologically essential. Tightness is a generalization of convexity, and the tightness of a triangulation is a fairly restrictive property. We give a review on all known examples of tight triangulations and formulate a (computer-aided) enumeration theorem for the case of at most 15 vertices and the presence of a vertex-transitive automorphism group. Altogether, six new examples of tight triangulations are presented, a vertex-transitive triangulation of the simply connected homogeneous 5-manifold SU(3)/SO(3) with vertex-transitive action, two non-symmetric 12-vertex triangulations of S3 × S2, and two non-symmetric triangulations of S3 × S3 on 13 vertices.

Enumeration and Random Realization of Triangulated Surfaces

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.

Small Examples of Nonconstructible Simplicial Balls and Spheres

We construct nonconstructible simplicial d-spheres with d + 10 vertices and nonconstructible, nonrealizable simplicial d-balls with d + 9 vertices for

One-point suspensions and wreath products of polytopes and spheres

d\geq 3

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

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Surface Realization with the Intersection Segment Functional

It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinational and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.

Examples of Z-Acyclic and Contractible Vertex-Homogeneous Simplicial Complexes

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R for all vertex-minimal triangulations of the orientable surfaces of genus g = 3 and g = 4. Moreover, for the first time, examples of simplicial polyhedra in R of genus 5 with 12 vertices were obtained.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.

Some Results Related to the Evasiveness Conjecture

Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry from the theoretical, algorithmic, and practical points of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection segment functional. This heuristic was used to find geometric realizations in ℝ3 for all vertex-minimal triangulations of the orientable surfaces of genera g = 3 and g = 4. Moreover, for the first time, examples of simplicial polyhedra in ℝ3 of genus 5 with 12 vertices have been obtained.