Classifies fake surfaces up to complexity 5.
problem Classifying fake surfaces for low-dimensional topology.
method Derived properties of fake surfaces, classified up to complexity 5.
result Proved conjectures about fake surfaces up to complexity 5.
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169,341 papers · 148 categories
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48 results for “contractible polyhedra”
The paper finds infinitely many Mazur type manifolds and corks with shadow complexity one.
problem Finding manifolds and corks with specific shadow complexity.
method Using Turaev's shadow and contractible special polyhedra with one true vertex.
result The discovery of infinitely many Mazur type manifolds and corks with shadow complexity one.
Polyhedra collapse to subpolyhedra if they can be continuously shrunk onto them.
problem Characterizing when a polyhedron can be continuously shrunk onto a subpolyhedron.
method Piecewise-linear free deformation retraction and metric considerations.
result A polyhedron collapses to a subpolyhedron if and only if it admits a free deformation retraction onto that subpolyhedron.
Let S be a compact surface of genus >1, and g be a metric on S of constant curvature K\in\{-1,0,1\} with conical singularities of negative singular curvature. When K=1 we add the condition that the lengths of the contractible geodesics are >2π. We prove that there exists a convex polyhedral surface P in the Lorentzian …
We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n-1)-polyhedron. If the n-polyhedron is contractible and n\ne 3 (or n=3 and the Andrews-Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding. In contrast, there…
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of c…
Löbell polyhedra have small systoles and are quasi-arithmetic.
problem Finding compact hyperbolic polyhedra with small systoles.
method Elementary and conceptual means to observe systole behavior, number theoretic invariants to refine results.
result Löbell polyhedra give examples of closed hyperbolic 3-manifolds with arbitrarily small systole and are quasi-arithmetic.
Introduces Coxeter polyhedra in various geometries.
problem None explicitly stated, focuses on introducing concepts.
method Introduction of Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries.
result Fundamental theorems and descriptions of polyhedra and tessellations.
New hyperbolic polyhedra with angles and volumes calculated.
problem Finding new hyperbolic polyhedra with specific dihedral angles.
method Constructed a new sequence of hyperbolic polyhedra with angles and determined their volumes.
result Volumes of some constructed polyhedra determined.
The paper sets new limits on hyperbolic polyhedra volumes.
problem Finding upper bounds on volumes of hyperbolic polyhedra.
method Analyzes three types of polyhedra: ideal, compact with finite vertices, and finite volume with mixed vertices.
result Establishes new upper bounds for polyhedra volumes in hyperbolic space.
Weakly Inscribed Polyhedramath.MG
Study of polyhedra on a sphere in projective 3-space.
problem Characterizing polyhedra with vertices on a sphere.
method Purely combinatorial and linear programming approaches.
result Characterization of dihedral angles and hyperbolic-de Sitter structure.
We develop some basic Lipschitz homotopy technique and apply it to manifolds with finite asymptotic dimension. In particular we show that the Higson compactification of a uniformly contractible manifold is mod acyclic in the finite dimensional case. Then we give an alternative proof of the Higher Signature Novikov …
Study mutant pairs of hyperbolic polyhedra, focusing on commensurability.
problem Determine commensurability of mutant pairs of hyperbolic polyhedra.
method Introduce mutation concept, develop new techniques for non-cusped polyhedra.
result New techniques needed for studying mutant pairs of polyhedra.
The study sets limits on dihedral angles of large hyperbolic polyhedra.
problem Establishing bounds on dihedral angles of hyperbolic Coxeter polyhedra.
method Developed a constructive procedure for Coxeter polyhedra with prescribed dihedral angles.
result Classification of ADEG-polyhedra with specific dihedral angles and no disjoint facets.
Algorithm finds smallest right-angled hyperbolic polyhedra.
problem Finding the smallest volume right-angled hyperbolic polyhedra.
method Algorithm implemented on computer using Orb to compute volumes.
result First 825 polyhedra in the list of smallest volume right-angled hyperbolic polyhedra in dimension 3 have been determined.
Easy condition for local k-connectedness in inverse limits of polyhedra.
problem Local k-connectedness of inverse limits of polyhedra.
method Easy condition for local k-connectedness.
result Provided an easily verifiable condition.
The study finds upper bounds and computes volumes of ideal right-angled polyhedra in Lobachevsky space.
problem Finding upper bounds and computing volumes of ideal right-angled polyhedra in Lobachevsky space.
method Analyzing a class of right-angled polyhedra with vertices on the absolute, obtaining upper bounds on volumes, computing volumes for polyhedra with up to 23 faces, and introducing the class of polyhedra with isolated triangles.
result Minimum volumes are realized on antiprisms and twisted antiprisms, and the first 248 values of volumes are presented.
Study confirms conjecture on Coxeter polyhedra growth rates in hyperbolic 3-space.
problem Growth rates of reflection groups in hyperbolic Coxeter polyhedra.
method Proved conjecture for ideal Coxeter polyhedra in ; found smallest growth rate; explored correlations between volumes and growth rates.
result Growth rates of ideal Coxeter polyhedra in are always Perron numbers.
Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
The article explores toric spaces of regular polyhedra, highlighting rational and non-rational cases.
problem Exploring toric spaces associated with regular convex polyhedra.
method Symplectic and complex toric spaces associated with five regular convex polyhedra.
result The regular dodecahedron and icosahedron cannot be treated via standard toric geometry.
We review several results related to the characterization of polyhedra in hyperbolic 3-space. In particular we present Rivin's theorem that gives a characterization of compact convex hyperbolic polyhedra, and Hodgson's proof of the Adreev's theorem. We also review the analogous characterization of ideal polyhedra, and …
Improved bounds on ideal vertices in right-angled hyperbolic polyhedra.
problem Finding bounds on ideal vertices in hyperbolic polyhedra.
method Improved Nikulin's inequality and Nonaka's lower bound.
result Shorter proofs and improved bounds on ideal vertices.
Study calculates mass of special polyhedra in hyperbolic space.
problem Evaluating mass in hyperbolic geometry.
method Used upper half space model and special polyhedra.
result Evaluated mass functional on polyhedra.
Mass in relativity linked to polyhedra geometry.
problem Mass in general relativity.
method Riemannian polyhedra geometry.
result Mass connected to polyhedra geometry.
Flexible models of non-flexible polyhedra explained.
problem Non-flexible Siamese dipyramids behave like flexible ones.
method Simple mathematical method to explain model flexibility.
result Physical models of Siamese dipyramids are flexible.
Polyhedra volume conjecture supports Stoker conjecture weakly.
problem Proving the Stoker conjecture for polyhedra.
method Using an extension of Montcouquiol and Weiss' result on polyhedra angles as local coordinates.
result Volume Conjecture for polyhedra implies a weak version of the Stoker conjecture.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
Intermediate logic of all convex polyhedra is axiomatized.
problem Defining and axiomatizing intermediate logic for convex polyhedra.
method Using Jankov-Fine formulas, classical polyhedral geometry, and p-morphic images to establish completeness.
result A finite axiomatisation of PL for all convex polyhedra.
Study approximates Riemannian manifolds using polyhedra.
problem Understanding Tullio Regge's approximation theorem.
method Proof of Regge theorem using polyhedra approximation.
result Integral of scalar curvature approximated by polyhedral curvature.
Geometrically realized polyhedra from directed trees, including associahedra.
problem Understanding the structure of associative algebras with co-inner products.
method Geometric realization of polyhedra using directed planar trees.
result These polyhedra, including associahedra, are homeomorphic to balls.
Analyzes 2D polyhedra decompositions into PL collapsible subpolyhedra.
problem Characterizing 2D polyhedra that can be split into two PL collapsible parts.
method Examines the minimum size of PL collapsible covers and uses topological and geometric analysis.
result Special class of 2D polyhedra appears naturally, including all closed surfaces and complexes from one-relator presentations.
The paper solves three problems related to monostable polyhedra.
problem Three problems related to monostable polyhedra posed by Conway and Goldberg.
method General theorem describing approximations of smooth convex bodies by convex polyhedra in terms of static equilibrium points.
result Existence of a convex polyhedron with only one stable and one unstable point.
New families of hyperbolic polyhedra yield infinitely many unique reflection groups.
problem Understanding commensurability classes of compact Coxeter polyhedra in hyperbolic spaces.
method Analyzing families of compact Coxeter polyhedra constructed by Makarov.
result Proves infinitely many commensurability classes in 4- and 5-dimensional hyperbolic spaces.
New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
Improved algorithm learns convex polyhedra with margin.
problem Learning convex polyhedra with margin in the realizable PAC setting.
method Constructs a consistent polyhedron as an intersection of halfspaces with constant-size margins.
result Polyhedra can be learned efficiently with margin constraints.
Biplanar Foldingsmath.DG
New framework for folding polyhedra into two planes.
problem Creating polyhedra that can be folded into two planes.
method General framework for bifoldable polyhedra construction.
result Examples of infinite triply periodic and fractal bifoldable polyhedra.
In this article we establish the relation between the spines of 3-manifolds and the polyhedra with identified faces. We do this by showing that the spines of the closed, connected, orientable 3-manifolds can be presented through polyhedra with identified faces in a very natural way. We also prove the equivalence betwee…
Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flowmath.GT
Paper solves long-standing problem of infinite ideal polyhedra in hyperbolic space.
problem Characterize infinite ideal polyhedra in hyperbolic 3-space.
method Introduced combinatorial Ricci flow for infinite ideal circle patterns.
result Proved characterization of infinite ideal circle patterns under specific conditions.
We consider a natural question: "Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence?" and a strongly related problem of K. Borsuk (1967): "Is it true that two ANR's homotopy dominating each other have the same homotopy type?" The answer was earlier known to be positive for ma…
Mass in relativity linked to polyhedra geometry.
problem Understanding ADM mass in general relativity.
method Relating ADM mass to the total mean curvature and defect of dihedral angles of Riemannian polyhedra.
result Expressed -dimensional mass as an integral of geometric quantities.
New periodic polyhedra found in curved spaces.
problem Existence of periodic polyhedra in curved spaces.
method Using Archimedean solids and transformations, constructing polyhedra with specific properties.
result Existence of compact polyhedral surfaces in spaceforms.
Notes on Shapes of Polyhedramath.GT
Course notes on polyhedra shapes from a graduate course.
problem Understanding polyhedra shapes from Thurston's complex proofs.
method Rewriting and simplifying Thurston's proofs.
result Course notes are useful for understanding polyhedra shapes.
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
Study of hyperideal polyhedra in anti-de Sitter space.
problem Characterizing hyperideal polyhedra in anti-de Sitter space.
method Defined hyperideal polyhedra as intersections with convex polyhedra in projective model of anti-de Sitter space.
result Hyperideal polyhedra uniquely determined by combinatorics, dihedral angles, and induced metrics on boundary.
Software finds ideal polyhedra with rational dihedral angles and volume maxima.
problem Finding ideal convex polyhedra with maximal volume in hyperbolic 3-space.
method Rivin's variational characterization and combinatorial optimization algorithms.
result Maximal volume ideal polyhedra have dihedral angles that are rational multiples of π.
The paper proves rigidity and uniformization theorems for infinite circle patterns and convex polyhedra in hyperbolic 3-space.
problem Characterize infinite circle patterns and convex polyhedra in hyperbolic 3-space.
method Extends techniques from previous work to prove rigidity and uniformization theorems for infinite circle patterns and convex polyhedra.
result Establishes existence and rigidity of infinite regular circle patterns and convex trivalent polyhedra.
Polyhedra rigidity theorem in hyperbolic space proved.
problem Dihedral rigidity conjecture in hyperbolic 3-space.
method Comparison theorem for polyhedra in a 3-manifold with scalar curvature bounded below.
result Confirms Gromov dihedral rigidity conjecture in hyperbolic 3-space.
New bounds found for vertices of hyperbolic polyhedra in dimensions 5 to 12.
problem Determining minimum number of ideal and finite vertices in hyperbolic polyhedra.
method Geometric method of orthogonal gluings combined with double counting and recurrence relations.
result Improved lower bounds for vertices in all dimensions up to 12.