Delaunay triangulation fails on dense point sets in Riemannian manifolds.
problem Obtaining a Delaunay triangulation for dense point sets on Riemannian manifolds.
method Analysis of Delaunay complexes on Riemannian manifolds, focusing on sample density.
result Sample density alone is not sufficient to ensure Delaunay triangulation in manifolds of dimension > 2.
Connected flip graphs for triangulations on hyperbolic surfaces.
problem Connecting triangulations on hyperbolic surfaces via flips.
method Proving connectedness of flip graphs and giving bounds on edge flips.
result Flip graphs of geometric triangulations are connected.
Tight triangulated manifolds are generalisations of neighborly triangulations of closed surfaces and are interesting objects in Combinatorial Topology. Tight triangulated manifolds are conjectured to be minimal. Except few, all the known tight triangulated manifolds are stacked. It is known that locally stacked tight t…
New isolated geometric triangulations found in once-punctured torus bundles.
problem Identifying isolated geometric triangulations in 3-manifolds.
method Examining ideal triangulations and their moves to find isolated geometric ones.
result Infinite family of once-punctured torus bundles with isolated geometric triangulations.
Efficient triangulations help in understanding 3-manifold boundaries.
problem Understanding boundary slopes in 3-manifolds.
method Introducing and studying boundary-efficient triangulations and inflating ideal triangulations.
result There are only finitely many boundary slopes for incompressible and \(\partial\)-incompressible surfaces in compact 3-manifolds.
A 6-regular triangulation for hyperbolic plane created.
problem Creating a 6-regular triangulation for hyperbolic plane.
method Constructed a 6-regular geodesic triangulation.
result A 6-regular geodesic triangulation of the hyperbolic plane was successfully created.
The study proves poor ideal three-edge triangulations are minimal for certain 3-manifolds.
problem Finding minimal ideal triangulations for specific 3-manifolds.
method Analyzing properties of poor ideal three-edge triangulations and applying them to construct minimal triangulations.
result Poor ideal three-edge triangulations are proven to be minimal for certain 3-manifolds.
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
Geometric triangulations can be transformed by bistellar moves.
problem Transforming geometric triangulations of different manifolds.
method Using bistellar moves, a type of local change to triangulations.
result Geometric triangulations of compact manifolds can be connected by bistellar moves.
Minimal triangulations for 229 hyperbolic census knots discovered.
problem Finding minimal triangulations for hyperbolic census knots.
method Ideal triangulations of the magic manifold, low-complexity triangulations for partial fillings, sorting into families.
result Minimal triangulations for 229 hyperbolic census knots discovered, conjectured to be minimal for all 42 families.
A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In…
Agol recently introduced the concept of a veering taut triangulation, which is a taut triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, givin…
Authors find small triangulations for specific 4-manifolds.
problem Finding optimal triangulations for 4-manifolds.
method Triangulated connected sums of CP^2 and S^2×S^2, conjectured minimal pentachora.
result Triangulations have the smallest number of pentachora for their types.
Combinatorial description of 3-manifolds using ordered triangulations.
problem Understanding closed 3-manifolds through ideal triangulations.
method Combining ordered ideal triangulations and Pachner moves.
result Closed 3-manifolds can be described via ordered triangulations and moves.
Minimal ideal triangulations studied for hyperbolic 3-manifolds.
problem Finding minimal triangulations of hyperbolic 3-manifolds.
method Characterization of low degree edges, layered solid torus subcomplexes, and 1-dimensional cohomology.
result Monodromy ideal triangulations of once-punctured torus bundles are minimal.
The paper constructs triangulations for double twist knots using geometric methods.
problem Constructing explicit triangulations of double twist knots.
method Using triangulating Dehn fillings, layered solid tori, and their double covers.
result Proves both triangulations are geometric, using conjecturally minimal triangulation to present A-polynomial equations.
New bounds show triangulated surfaces are evenly distributed in moduli space.
problem Distribution of triangulated surfaces in moduli space as genus increases.
method Proved upper and lower bounds for the number of triangulated surfaces in Teichmüller balls.
result Number of triangulated surfaces in a Teichmüller unit ball is at most exponential in the number of triangles, independent of genus.
Experimental results on veering triangulations of 3-manifolds.
problem Understanding the combinatorial structure of veering triangulations.
method Algorithmic construction and experimental analysis.
result Experimental insights into the structure of veering triangulations and their relation to topological invariants.
We investigate a type of distance between triangulations on finite type surfaces where one moves between triangulations by performing simultaneous flips. We consider triangulations up to homeomorphism and our main results are upper bounds on distance between triangulations that only depend on the topology of the surfac…
Researchers found the minimum number of tetrahedra needed to triangulate elliptic and sol 3-manifolds.
problem Finding the minimum number of tetrahedra in triangulations of 3-manifolds.
method Computed the triangulation complexity of all elliptic and sol 3-manifolds, within a bounded error.
result Computed the triangulation complexity of all elliptic and sol 3-manifolds.
With the [0,1,2]-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic q-equivelar triangulations of orientable and non-orientable surfaces for every q=3k, k≥2, and every q=3k+1, k≥3. Series of cy…
New loom spaces link flows and triangulations.
problem Understanding flows and triangulations in 3D.
method Introducing loom spaces and proving associated triangulations.
result Locally veering triangulations can be associated to loom spaces.
New method connects veering triangulations to dynamic pairs.
problem Understanding veering triangulations and their properties.
method Shearing decomposition of veering triangulations.
result Canonically associated dynamic pairs of branched surfaces.
New triangulations encode flows with vanishing polynomial.
problem Constructing veering triangulations with vanishing taut polynomial.
method Using connections between veering triangulations and pseudo-Anosov flows.
result Created arbitrarily large veering triangulations with vanishing taut polynomial.
Proofs contractibility of geodesic triangulations spaces and non-trivial homotopy groups.
problem Contractibility and homotopy groups of geodesic triangulations.
method Short proofs and existence proofs for specific cases.
result Existence of polygon triangulations with non-trivial nth homotopy groups.
This paper derives formulas for Chern classes of triangulated circle bundles using combinatorial necklaces.
problem Calculating Chern classes for triangulated circle bundles over polyhedra.
method Using triangulations and necklace combinatorics, the paper derives rational parity formulas for Chern classes.
result Rational parity formulas for Chern classes of triangulated circle bundles are derived.
0-efficient triangulations of 3-manifolds are defined and studied. It is shown that any triangulation of a closed, orientable, irreducible 3-manifold M can be modified to a 0-efficient triangulation or M can be shown to be one of the manifolds S^3, RP^3 or L(3,1). Similarly, any triangulation of a compact, orientable, …
The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its "minimal layered triangulation." This paper proves that for each integer n>1…
The study proves which circle bundles can be triangulated over a 3-simplex boundary.
problem Which circle bundles can be triangulated over a 3-simplex boundary?
method Proof of which circle bundles can be triangulated over a 3-simplex boundary.
result Only trivial and Hopf circle bundles can be triangulated over a 3-simplex boundary.
Software simplifies triangulations of 4-manifolds, revealing exotic structures.
problem Understanding smooth 4-manifolds from discrete and algorithmic perspectives.
method New software tools, including an algorithm for triangulations from Kirby diagrams and a heuristic for simplification.
result Presented new triangulations of exotic pairs, corks, and plugs, including the smallest known K3 surface.
Minimal Delaunay triangulations on hyperbolic surfaces have linear number of vertices.
problem Finding the minimum number of vertices in Delaunay triangulations of hyperbolic surfaces.
method Analyzing the genus g of hyperbolic surfaces to derive bounds on the number of vertices. result The number of vertices in minimal Delaunay triangulations of hyperbolic surfaces is linear in the genus g. Essential triangulations of certain manifolds are connected via specific moves.
problem Connecting essential triangulations of certain manifolds.
method Essential triangulations are connected via 2-3 and 3-2 moves alone, ignoring those for which no 2-3 move preserves essentiality.
result Essential triangulations of certain manifolds are connected via 2-3 and 3-2 moves alone.
New triangulations for twist knots, proving volume conjecture.
problem Computing the volume of twist knot complements.
method Ideal triangulations and H-triangulations, using Thurston's method and volume functional.
result Proved the Teichmüller TQFT volume conjecture for all twist knots.
Proving geodesic triangulation spaces are Euclidean.
problem Proving spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
method Proposing an approach to prove homeomorphism using negative curvature surfaces.
result Spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
New triangulations of quaternionic projective plane found with various symmetry groups.
problem Classifying triangulations of quaternionic projective plane with 15 vertices.
method Constructing and classifying 15-vertex triangulations with various symmetry groups.
result Exactly 75 triangulations of quaternionic projective plane with 15 vertices and symmetry group of order at least 4.
Method samples triangulations of manifolds using biased random walks.
problem Efficiently sample triangulations of manifolds.
method Biased random walk through Pachner graph with Metropolis-Hastings accept/reject probabilities.
result Samples triangulations at random from chosen probability, estimating rare triangulations.
Flips connect all triangulations of flat surfaces.
problem Connecting triangulations of flat surfaces with flips.
method Proving any two triangulations can be linked by flips.
result Flips can connect all triangulations of flat surfaces.
We study several properties of $\ZZ_2^n$-equivariant triangulations of $\RR P^n$. We show that a $\ZZ_2^n$-equivariant triangulation of $\RR P^n$ induces a triangulated subdivision of the orbit space △n. We show that any vertex minimum $\ZZ_2^3$-equivariant triangulation of $\RR P^3$ contains 11 verti…
Algorithm checks if geometrically triangulated manifolds are isometric.
problem Determining if two geometric triangulations of manifolds are isometric.
method Sequence of Pachner moves and barycentric subdivisions with bounds on lengths.
result Bounding the length of transformations between triangulations.
The paper finds canonical triangulations for specific 3-manifolds.
problem Finding canonical decompositions for cusped hyperbolic 3-manifolds.
method Showed local convexity at every face of the geometric triangulation.
result Found canonical triangulations for Dehn fillings of the Borromean rings link complement and related manifolds.
It is conjectured that every cusped hyperbolic 3-manifold has a decomposition into positive volume ideal hyperbolic tetrahedra (a "geometric" triangulation of the manifold). Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a ne…
Legendrian arcs connect veering triangulations to Anosov flows.
problem Connecting veering triangulations to Anosov flows for study.
method Realizing edges as Legendrian arcs with a bicontact structure.
result Veering triangulations can be placed in steady position.
This is the second in a series of papers in which we investigate ideal triangulations of the interiors of compact 3-manifolds with tori or Klein bottle boundaries. Such triangulations have been used with great effect, following the pioneering work of Thurston. Ideal triangulations are the basis of the computer program …
Study on veering triangulations and their flow graphs, proving new applications.
problem Understanding the structure of veering triangulations and their flow graphs.
method Analyzing the infinitesimal components of the flow graph associated with veering triangulations.
result Infinitesimal components of veering triangulations' flow graphs have specific forms related to subsets called 'walls'.
Geometric triangulations of surfaces with uniform vertex degree are shown.
problem Characterizing triangulations of surfaces with uniform vertex degree.
method Combination of combinatorial topology and induction to prove uniqueness of triangulations.
result Any degree-regular triangulation of a surface is geometric.
Infinite type surfaces can be perfectly divided into triangles.
problem Triangulating surfaces of infinite type.
method Showed arcs can be completed into triangulations if they intersect curves a finite number of times.
result Any surface of infinite type admits an ideal triangulation.
Bounding shears in ideal triangulations on hyperbolic surfaces.
problem Bounding shears in ideal triangulations on hyperbolic surfaces.
method Showing an ideal triangulation with bounded shear parameters on hyperbolic surfaces.
result An upper bound on shear parameters depends logarithmically on the surface's topology.
Minimal triangulations of circle bundles linked to circular permutations.
problem Which circle bundles can be triangulated over a given base triangulation?
method Minimal triangulations encoded by local systems of circular permutations of vertices.
result Classical Huntington transitivity axiom for cyclic orders expressed as a binary Chern cocycle.