Proves ideal triangulations of hyperbolic alternating links are non-degenerate.
problem Proving ideal triangulations of hyperbolic alternating links are non-degenerate.
method Using non-positively curved cubings of prime alternating link exteriors.
result Ideal triangulations are non-degenerate, guaranteeing hyperbolicity equations have solutions.
Study on polyhedra rigidity, finding non-existence of flexible weakly convex decomposable polyhedra.
problem Proving all decomposable polyhedra with vertices in convex position are infinitesimally rigid.
method Constructing explicit families of polyhedra, using the Hessian of the discrete Hilbert-Einstein functional, and searching for eigenvalues of the Hessian with Mathematica.
result Experimental evidence suggests no flexible, weakly convex and decomposable polyhedra exist.
Enhances understanding of stability conditions on surfaces.
problem Understanding stability conditions on surfaces.
method Introduces cluster exchange groupoid and uses triangulation covering graphs.
result Space of stability conditions is simply connected.
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension n, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…
Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
problem Embedding maps in higher dimensions without self-intersections.
method Lifting maps to embeddings in product spaces.
result Maps can be embedded in higher dimensions if they lift to embeddings in product spaces.
The paper constructs simplicial maps of any degree on spheres, solving a long-standing problem.
problem Constructing simplicial maps of any degree on spheres.
method Using connected sums and facet orientations, the paper develops a method to construct maps of any prescribed degree.
result The paper answers a question posed by Ryabichev and constructs simplicial maps of degree d for large d. Classifies homogeneous Levi non-degenerate hypersurfaces in complex 3-space.
problem Classifying specific types of complex hypersurfaces.
method Analyzing hypersurfaces with symmetry algebra of dimension at least 6.
result All such hypersurfaces are classified.
The punctured solenoid § is an initial object for the category of punctured surfaces with morphisms given by finite covers branched only over the punctures. The (decorated) Teichmüller space of § is introduced, studied, and found to be parametrized by certain coordinates on a fixed triangulation of §. Furthermore…
Study on hypersurfaces with minimized distance between rulings.
problem Characterizing singularities and properties of two-ruled hypersurfaces.
method Characterization through striction curves and examination of pseudo-non-degenerate properties.
result Two-ruled hypersurfaces constructed from specific curves are pseudo-non-degenerate.
New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
Theorem shows generic metrics yield non-degenerate geodesic nets.
problem Characterizing geodesic nets on generic metrics.
method Proving all connected embedded nets are non-degenerate for Baire-generic metrics.
result All stationary geodesic nets are non-degenerate for generic metrics.
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 move proves infinitely many non-conjugate braids for certain links.
problem Proving infinitely many non-conjugate braids for certain link types.
method Introducing a non-degenerate exchange move.
result Links have infinitely many non-conjugate braids after applying the move.
Develops new Poisson structures for moduli spaces.
problem Creating Poisson structures for moduli spaces.
method Introduces quasi Poisson and quasi Hamiltonian structures, novel momentum mappings.
result Bijective correspondence between quasi Poisson and quasi Hamiltonian structures.
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.
We show that non-degenerate hyperquadrics in R^{n+2} admit no skew branes. Stated more traditionally, a compact codimension-one immersed submanifold of a non-degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points. Similar results have been proven by others, but (except for e…
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.
The study resolves a conjecture about harmonic forms on compact manifolds.
problem Finding non-degenerate Z2-harmonic 1-forms on compact manifolds. method Develops a gluing theorem for non-degenerate Z2-harmonic 1-forms on compact manifolds. result Proves the existence of non-degenerate Z2-harmonic 1-forms on compact manifolds with positive first Betti number. 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.
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
Triangulations without degree one edges are connected via moves.
problem Connectivity of triangulations without degree one edges.
method 2-3 and 3-2 moves.
result Subgraph of Pachner graph without degree one edges is connected.
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…
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.
The paper classifies almost complex manifolds with non-degenerate torsion and their coverings.
problem Classification of almost complex manifolds with non-degenerate torsion.
method Analysis of homogeneous and non-homogeneous manifolds, classification of Lie algebras, covering spaces.
result Classification of manifolds with solvable Lie algebra up to coverings.
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…
Smooth structures enable solving PDEs via optimized triangulations.
problem Solving partial differential equations (PDEs) on complex spaces.
method Introducing Frölicher space structure on CW complexes and spaces of triangulations to enable differential methods.
result Optimized triangulations can be used to solve standard PDEs.
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.
The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.
problem Invariance and structure preservation under conformal-projective transformations.
method Proof of invariance and preservation of structures under conformal-projective transformations.
result Semi-Weyl and statistical structures with torsion are invariant under conformal-projective transformations.
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.
New findings on non-congruent curves with identical signatures.
problem Identifying congruence of non-degenerate curves with non-simple signatures.
method Associated directed graphs to signatures and used paths to reflect global and local symmetries.
result Non-congruent, non-degenerate curves can have identical signatures.
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, …
Researchers create non-degenerate harmonic functions on n-dimensional space.
problem Creating non-degenerate Z2-harmonic functions on Rn. method Using a variant of ellipsoidal coordinates, the construction is explicit and involves Lawlor's necks in Cn. result First known family of non-degenerate Z2-harmonic 1-forms with compact branching sets.