Triangulations Of Manifolds Research Articles

Statistical mechanics of graph models and their implications for emergent spacetime manifolds

Inspired by "quantum graphity" models for spacetime, a statistical model of graphs is proposed to explore possible realizations of emergent manifolds. Graphs with given numbers of vertices and edges are considered, governed by a very general Hamiltonian that merely favors graphs with near-constant valency and local rotational symmetry. The ratio of vertices to edges controls the dimensionality of the emergent manifold. The model is simulated numerically in the canonical ensemble for a given vertex to edge ratio, where it is found that the low-energy states are almost triangulations of two-dimensional manifolds. The resulting manifold shows topological "handles" and surface intersections in a higher embedding space, as well as non-trivial fractal dimension consistent with previous spectral analysis, and nonlocal links consistent with models of disordered locality. The transition to an emergent manifold is first order, and thus dependent on microscopic structure. Issues involved in interpreting nearly-fixed valency graphs as Feynman diagrams dual to a triangulated manifold as in matrix models are discussed. Another interesting phenomenon is that the entropy of the graphs are super-extensive, a fact known since Erd\H{o}s, which results in a transition temperature of zero in the limit of infinite system size: infinite manifolds are always disordered. Aside from a finite universe or diverging coupling constraints as possible solutions to this problem, long-range interactions between vertex defects also resolve the problem and restore a nonzero transition temperature, in a manner similar to that in low-dimensional condensed-matter systems.

On r-stacked triangulated manifolds

The notion of $r$-stackedness for simplicial polytopes was introduced by McMullen and Walkup in 1971 as a generalization of stacked polytopes. In this paper, we define the $r$-stackedness for triangulated homology manifolds and study their basic properties. In addition, we find a new necessary condition for face vectors of triangulated manifolds when all the vertex links are polytopal. Généralisant les polytopes simpliciaux empilés, McMullen et Walkup ont introduit en 1971 la notion de $r$-empilement pour les polytopes simpliciaux. Dans cet article, nous définissons la notion de $r$-empilement pour les variétés homologiques simpliciales et étudions ses propriétés élémentaires. En outre, nous donnons une nouvelle condition pour les $f$-vecteurs des variétés simpliciales lorsque tous les sommets ont un lien polytopal.

A Wecken theorem for n-valued maps

In [6] Schirmer (1985) established that, if φ:X⊸X is an n-valued map defined on a compact triangulable manifold of dimension at least three, then the appropriate Nielsen number, N(φ), is a sharp lower bound for the number of fixed points in the n-valued homotopy class of φ. In this note we generalize this theorem by allowing X to be any compact polyhedron without local cut points and such that no connected component is a two-manifold.

Poincaré 2-group and quantum gravity

We show that general relativity can be formulated as a constrained topological theory for flat 2-connections associated with the Poincaré 2-group. Matter can be consistently coupled to gravity in this formulation. We also show that the edge lengths of the spacetime manifold triangulation arise as the basic variables in the path-integral quantization, while the state-sum amplitude is an evaluation of a colored 3-complex, in agreement with the category theory results. A 3-complex amplitude for Euclidean quantum gravity is proposed.

Dynamic programming and viscosity solutions for the optimal control of quantum spin systems

The purpose of this paper is to describe the application of the notion of viscosity solutions to solve the Hamilton–Jacobi–Bellman (HJB) equation associated with an important class of optimal control problems for quantum spin systems. The HJB equation that arises in the control problems of interest is a first-order nonlinear partial differential equation defined on a Lie group. Hence we employ recent extensions of the theory of viscosity solutions to Riemannian manifolds in order to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used develop a finite difference approximation method for numerically computing the solution to such problems. The convergence of these approximations is proven using viscosity solution methods. In order to illustrate the techniques developed, these methods are applied to an example problem.

Quading triangular meshes with certain topological constraints

In computer graphics and geometric modeling, shapes are often represented by triangular meshes (also called 3D meshes or manifold triangulations). The quadrangulation of a triangular mesh has wide applications. In this paper, we present a novel method of quading a closed orientable triangular mesh into a quasi-regular quadrangulation, i.e., a quadrangulation that only contains vertices of degree four or five. The quasi-regular quadrangulation produced by our method also has the property that the number of quads of the quadrangulation is the smallest among all the quasi-regular quadrangulations. In addition, by constructing the so-called orthogonal system of cycles our method is more effective to control the quality of the quadrangulation.

Complete Tensor Field Topology on 2D Triangulated Manifolds embedded in 3D

AbstractThis paper is concerned with the extraction of the surface topology of tensor fields on 2D triangulated manifolds embedded in 3D. In scientific visualization topology is a meaningful instrument to get a hold on the structure of a given dataset. Due to the discontinuity of tensor fields on a piecewise planar domain, standard topology extraction methods result in an incomplete topological skeleton. In particular with regard to the high computational costs of the extraction this is not satisfactory. This paper provides a method for topology extraction of tensor fields that leads to complete results. The core idea is to include the locations of discontinuity into the topological analysis. For this purpose the model of continuous transition bridges is introduced, which allows to capture the entire topology on the discontinuous field. The proposed method is applied to piecewise linear three‐dimensional tensor fields defined on the vertices of the triangulation and for piecewise constant two or three‐dimensional tensor fields given per triangle, e.g. rate of strain tensors of piecewise linear flow fields.

Coverings and minimal triangulations of 3–manifolds

This paper uses results on the classification of minimal triangulations of 3-manifolds to produce additional results, using covering spaces. Using previous work on minimal triangulations of lens spaces, it is shown that the lens space $L(4k, 2k-1)$ and the generalised quaternionic space $S^3/Q_{4k}$ have complexity $k,$ where $k\ge 2.$ Moreover, it is shown that their minimal triangulations are unique.

On Walkup’s class [formula omitted] and a minimal triangulation of [formula omitted

For d ≥ 2 , Walkup’s class K ( d ) consists of the d -dimensional simplicial complexes all whose vertex-links are stacked ( d − 1 ) -spheres. Kalai showed that for d ≥ 4 , all connected members of K ( d ) are obtained from stacked d -spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f 1 ≥ 5 f 0 − 15 2 χ , with equality only for X ∈ K ( 4 ) . Kühnel observed that this implies f 0 ( f 0 − 11 ) ≥ − 15 χ , with equality only for 2-neighborly members of K ( 4 ) . Kühnel also asked if there is a triangulated 4-manifold with f 0 = 15 , χ = − 4 (attaining equality in his lower bound). In this paper, guided by Kalai’s theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S 3 ▪ S 1 . Because of Kühnel’s inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of ( 2 d + 3 ) -vertex sphere products S d − 1 × S 1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai’s result.

Stacked polytopes and tight triangulations of manifolds

Tightness of a triangulated manifold is a topological condition, roughly meaning that any simplex-wise linear embedding of the triangulation into Euclidean space is “as convex as possible”. It can thus be understood as a generalization of the concept of convexity. In even dimensions, super-neighborliness is known to be a purely combinatorial condition which implies the tightness of a triangulation. Here, we present other sufficient and purely combinatorial conditions which can be applied to the odd-dimensional case as well. One of the conditions is that all vertex links are stacked spheres, which implies that the triangulation is in Walkupʼs class K ( d ) . We show that in any dimension d ⩾ 4 , tight-neighborly triangulations as defined by Lutz, Sulanke and Swartz are tight. Furthermore, triangulations with k-stacked vertex links and the centrally symmetric case are discussed.

Simpcomp

simpcomp [4] is an extension (a so called package) to GAP [6], the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of manifolds. For an introduction to the field of piecewise linear (PL) topology see the books [16] and [11]. The authors acknowledge support by the DFG: simpcomp has partly been developed within the DFG projects Ku 1203/5-2 and Ku 1203/5-3.

Interpolation of Geodesic B-spline Curves on Manifold Triangulation

In allusion to the deficiencies of the existing methods of designing free curve on the surface,a generation method for geodesic B-spline curve interpolated from manifold triangulation surface is proposed.The shortest geodesic curve between two points on the manifold triangulation surface is used to substitute for the straight line between the two points in Euclidean space;and de Boor algorithm in Euclidean space is expand to curved space,so that the representation for geodesic B-spline curve on manifold triangulation is obtained.The given range of points constrained on the mesh surface are operated base on the B-spline interpolation theory in Euclidean space;control points are gained by inverse computation,and then projected onto the mesh surface;points on the mesh surface are treated as the initial control points of expected curve,from which the initial geodesic B-spline curve is finally generated.A reverse error compensation strategy which obtains control points constrained onto the mesh by simple iteration is proposed in order to approximate the curve with the data points as much as possible.According to the convex hull property of the curve,the convex hull region is virtual partitioned from the whole mesh surface;and then the computation of interpolating geodesic B-spline curve is only related to the mesh vertices within the convex hull region rather than the overall mesh,which greatly reduces the computational complexity.The results show that,the method is robust,effective,and able to meet the requirements of curve interaction design on surface.

Flag-no-square triangulations and Gromov boundaries in dimension 3

We describe an infinite family of 3-dimensional topological spaces, which are homeomorphic to boundaries of certain word-hyperbolic groups. The groups are right-angled hyperbolic Coxeter groups, whose nerves are flag-no-square triangulations of 3-dimensional manifolds. We prove that any 3-dimensional polyhedral complex (in particular, any 3-manifold) can be triangulated in a flag-no-square way.

Cognitive Behavioral Therapy, Sertraline, and Combination Effective for Childhood Anxiety Disorders

For d≥2, Walkup’s class K(d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d−1)-spheres. Kalai showed that for d≥4, all connected members of K(d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic χ satisfies f1≥5f0−152χ, with equality only for X∈K(4). Kühnel observed that this implies f0(f0−11)≥−15χ, with equality only for 2-neighborly members of K(4). Kühnel also asked if there is a triangulated 4-manifold with f0=15, χ=−4 (attaining equality in his lower bound). In this paper, guided by Kalai’s theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S3S1. Because of Kühnel’s inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d+3)-vertex sphere products Sd−1×S1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai’s result.

BRAIDING AND ENTANGLEMENT IN SPIN NETWORKS: A COMBINATORIAL APPROACH TO TOPOLOGICAL PHASES

The spin network quantum simulator relies on the su(2) representation ring (or its q-deformed counterpart at q = root of unity) and its basic features naturally include (multipartite) entanglement and braiding. In particular, q-deformed spin network automata are able to perform efficiently approximate calculations of topological invarians of knots and 3-manifolds. The same algebraic background is shared by 2D lattice models supporting topological phases of matter that have recently gained much interest in condensed matter physics. These developments are motivated by the possibility to store quantum information fault-tolerantly in a physical system supporting fractional statistics since a part of the associated Hilbert space is insensitive to local perturbations. Most of currently addressed approaches are framed within a "double" quantum Chern–Simons field theory, whose quantum amplitudes represent evolution histories of local lattice degrees of freedom.We propose here a novel combinatorial approach based on "state sum" models of the Turaev–Viro type associated with SU(2)q-colored triangulations of the ambient 3-manifolds. We argue that boundary 2D lattice models (as well as observables in the form of colored graphs satisfying braiding relations) could be consistently addressed. This is supported by the proof that the Hamiltonian of the Levin–Wen condensed string net model in a surface Σ coincides with the corresponding Turaev–Viro amplitude on Σ × [0,1] presented in the last section.

Geometric torsions and invariants of manifolds with a triangulated boundary

Geometric torsions are torsions of acyclic complexes of vector spaces consisting of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a three-dimensional manifold with a triangulated boundary. These invariants can be naturally combined into a vector, and a change of the boundary triangulation corresponds to a linear transformation of this vector. Moreover, when two manifolds are glued at their common boundary, these vectors undergo scalar multiplication, i.e., they satisfy Atiyah’s axioms of a topological quantum field theory.

Modeling on triangulations with geodesic curves

This paper discusses the problem of modeling on triangulated surfaces with geodesic curves. In the first part of the paper we define a new class of curves, called geodesic Bezier curves, that are suitable for modeling on manifold triangulations. As a natural generalization of Bezier curves, the new curves are as smooth as possible. In the second part we discuss the construction of C 0 and C 1 piecewise Bezier splines. We also describe how to perform editing operations, such as trimming, using these curves. Special care is taken to achieve interactive rates for modeling tasks. The third part is devoted to the definition and study of convex sets on triangulated surfaces. We derive the convex hull property of geodesic Bezier curves.

Random tilings of spherical 3-manifolds

A set of random tilings for the compact Euclidean 3-manifolds have been considered recently. In this paper, non-deterministic triangulations of spherical 3-manifolds based on recursive procedures are introduced. Simplicial structures for lens, prism and Poincaré spherical dodecahedron spaces are described. The configurational entropies for the random structures depend only on the information in 2D and are consistent with the Bekenstein–Hawking bound.

An A ∞ structure on simplicial complexes

We consider a discrete (finite-difference) analogue of differential forms defined on simplicial complexes, in particular, on triangulations of smooth manifolds. Various operations are explicitly defined on these forms including the exterior differential d and the exterior product ∧. The exterior product is nonassociative but satisfies a more general relation, the so-called A ∞ structure. This structure includes an infinite set of operations constrained by the nilpotency relation (d + ∧ + m + …)n = 0 of the second degree, n = 2.

A formula for the Euler characteristics of even dimensional triangulated manifolds

An alternative formula for the Euler characteristics of even dimengulated manifolds is deduced from the generalized Dehn-Sommersional triangulated manifolds is deduced from the generalized Dehn-Sommerville equations.