Triangulations Of Manifolds Research Articles

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Circle-valued angle structures and obstruction theory

We study spaces of circle-valued angle structures, introduced by Feng Luo, on ideal triangulations of 3-manifolds. We prove that the connected components of these spaces are enumerated by certain cohomology groups of the 3-manifold with [Formula: see text]-coefficients. Our main theorem shows that this establishes a geometrically natural bijection between the connected components of the spaces of circle-valued angle structures and the obstruction classes to lifting boundary-parabolic PSL[Formula: see text]-representations of the fundamental group of the 3-manifold to boundary-unipotent representations into SL[Formula: see text]. In particular, these connected components have topological and algebraic significance independent of the ideal triangulations chosen to construct them. The motivation and main application of this study is to understand the domain of the state-integral defining the meromorphic 3D-index of Garoufalidis and Kashaev, necessary in order to classify the boundary-parabolic representations contributing to its asymptotics.

Simplicial volume of manifolds with amenable fundamental group at infinity

Abstract We show that for $n \neq 1,4$ , the simplicial volume of an inward tame triangulable open $n$ -manifold $M$ with amenable fundamental group at infinity at each end is finite; moreover, we show that if also $\pi _1(M)$ is amenable, then the simplicial volume of $M$ vanishes. We show that the same result holds for finitely-many-ended triangulable manifolds which are simply connected at infinity.

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

In 1987 Brehm and Kühnel showed that any combinatorial $d$-manifold with less than $3d/2+3$ vertices is PL homeomorphic to the sphere and any combinatorial $d$-manifold with exactly $3d/2+3$ vertices is PL homeomorphic to either the sphere or a manifold like a projective plane in the sense of Eells and Kuiper. The latter possibility may occur for $d\in\{2,4,8,16\}$ only. There exist a unique $6$-vertex triangulation of $\mathbb{RP}^2$, a unique $9$-vertex triangulation of $\mathbb{CP}^2$, and at least three $15$-vertex triangulations of $\mathbb{HP}^2$. However, until now, the question of whether there exists a $27$-vertex triangulation of a manifold like the octonionic projective plane has remained open. We solve this problem by constructing a lot of examples of such triangulations. Namely, we construct $634$ vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Four of them have symmetry group $\mathrm{C}_3^3\rtimes \mathrm{C}_{13}$ of order $351$, and the other $630$ have symmetry group $\mathrm{C}_3^3$ of order $27$. Further, we construct more than $10^{103}$ non-vertex-transitive $27$-vertex combinatorial $16$-manifolds like the octonionic projective plane. Most of them have trivial symmetry group, but there are also symmetry groups $\mathrm{C}_3$, $\mathrm{C}_3^2$, and $\mathrm{C}_{13}$. We conjecture that all the triangulations constructed are PL homeomorphic to the octonionic projective plane $\mathbb{OP}^2$. Nevertheless, we have no proof of this fact so far.

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

634 vertex-transitive and more than $10^{103}$ non-vertex-transitive 27-vertex triangulations of manifolds like the octonionic projective plane

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

On possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane

Tight Complexes Are Golod

Abstract Golodness of a simplicial complex is defined algebraically in terms of the Stanley–Reisner ring, and it has been a long-standing problem to find its combinatorial characterization. Tightness of a simplicial complex is a combinatorial analogue of a tight embedding of a manifold into Euclidean space, and has been studied in connection to minimal manifold triangulations. In this paper, by using an idea from toric topology, we prove that tight complexes are Golod, and as a corollary, we obtain that a triangulation of a closed connected orientable manifold is Golod if and only if it is tight, which is a combinatorial characterization of Golodness for manifold triangulations.

Triangulations of grassmannians and flag manifolds

MacPherson (in: Topological methods in modern mathematics: a symposium in Honor of John Milnor’s Sixtieth Birthday Stony Brook NY 1991, Perish, Houston, 1993. https://doi.org/10.2307/1970177) conjectured that the Grassmannian textrm{Gr}(2, mathbb {R}^n) has the same homeomorphism type as the combinatorial Grassmannian text{ MacP }(2,n), while Babson (in: A combinatorial flag space, MIT, 1993. https://doi.org/10.2307/1970177) proved that the spaces text{ Gr }(2,mathbb {R}^n) and text{ Gr }(1,2,mathbb {R}^n) are homotopy equivalent to their combinatorial analogs, the simplicial complexes Vert text{ MacP }(2,n)Vert and Vert text{ MacP }(1,2,n)Vert respectively. We will prove that text{ Gr }(2, mathbb {R}^n) and text{ Gr }(1,2, mathbb {R}^n) are homeomorphic to Vert text{ MacP }(2,n)Vert and Vert text{ MacP }(1,2,n)Vert respectively.

Shellable tilings on relative simplicial complexes and their h-vectors

Abstract An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h-tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h-tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h-vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

Algebraic h-vectors of simplicial complexes through local cohomology, part 1

Given an infinite field k and a simplicial complex Δ, a common theme in studying the f- and h-vectors of Δ has been the consideration of the Hilbert series of the Stanley–Reisner ring k[Δ] modulo a generic linear system of parameters Θ. Historically, these computations have been restricted to special classes of complexes (most typically triangulations of spheres or manifolds). We provide a compact topological expression of hd−1a(Δ), the dimension over k in degree d−1 of k[Δ]/(Θ), for any complex Δ of dimension d−1. In the process, we provide tools and techniques for the possible extension to other coefficients in the Hilbert series.

Restricted Delaunay Triangulation for Explicit Surface Reconstruction

The task of explicit surface reconstruction is to generate a surface mesh by interpolating a given point cloud. Explicit surface reconstruction is necessary when the point cloud is required to appear exactly on the surface. However, for a non-perfect input, such as lack of normals, low density, irregular distribution, thin and tiny parts, and high genus, a robust explicit reconstruction method that can generate a high-quality manifold triangulation is missing. We propose a robust explicit surface reconstruction method that starts from an initial simple surface mesh, alternately performs a Filmsticking step and a Sculpting step of the initial mesh, and converges when the surface mesh interpolates all input points (except outliers) and remains stable. The Filmsticking is to minimize the geometric distance between the surface mesh and the point cloud through iteratively performing a restricted Voronoi diagram technique on the surface mesh, whereas the Sculpting is to bootstrap the Filmsticking iteration from local minima by applying appropriate geometric and topological changes of the surface mesh. Our algorithm is fully automatic and produces high-quality surface meshes for non-perfect inputs that are typically considered to be challenging for prior state of the art. We conducted extensive experiments on simulated scans and real scans to validate the effectiveness of our approach.

Period collapse in Ehrhart quasi-polynomials of \(\{1,3\}\)-graphs

A graph whose nodes have degree \(1\) or \(3\) is called a \(\{1,3\}\)-graph. Liu and Osserman associated a polytope to each \(\{1,3\}\)-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set \(\{0,\frac14,\frac12,1\}\), which implies that the period of their Ehrhart quasi-polynomials is either \(1, 2\), or \(4\). We show that the period of the Ehrhart quasi-polynomial of these polytopes is 2 if the graph is a tree, the period is at most 2 if the graph is cubic, and the period is \(4\) otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.Mathematics Subject Classifications: 05C30, 05C76, 52B20Keywords: Ehrhart polynomials, period collapse

Decompositions of quasirandom hypergraphs into hypergraphs of bounded degree

We prove that any quasirandom uniform hypergraph H H can be approximately decomposed into any collection of bounded degree hypergraphs with almost as many edges. In fact, our results also apply to multipartite hypergraphs and even to the sparse setting when the density of H H quickly tends to 0 0 in terms of the number of vertices of H H . Our results answer and address questions of Kim, Kühn, Osthus and Tyomkyn; and Glock, Kühn and Osthus as well as Keevash. The provided approximate decompositions exhibit strong quasirandom properties which are very useful for forthcoming applications. Our results also imply approximate solutions to natural hypergraph versions of long-standing graph decomposition problems, as well as several decomposition results for (quasi)random simplicial complexes into various more elementary simplicial complexes such as triangulations of spheres and other manifolds.

Knotted 4-regular graphs: Polynomial invariants and the Pachner moves

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs—one inspired by the Kauffman bracket relations and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.

On the Nielsen root theory of n-valued maps

Let phi :X multimap Y be an n-valued map of connected finite polyhedra and let a in Y. Then, x in X is a root of phi at a if a in phi (x). The Nielsen root number N(phi : a) is a lower bound for the number of roots at a of any n-valued map homotopic to phi . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given epsilon > 0, there is an n-valued map psi homotopic to phi within Hausdorff distance epsilon of phi such that psi has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, q ne 2, then there is an n-valued map homotopic to phi that has N(phi : a) roots at a. We verify the conjecture when X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.

The Topological Correctness of PL Approximations of Isomanifolds

Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate multivalued smooth function f: {mathbb {R}}^drightarrow {mathbb {R}}^{d-n}. A natural (and efficient) way to approximate an isomanifold is to consider its piecewise-linear (PL) approximation based on a triangulation mathcal {T} of the ambient space {mathbb {R}}^d. In this paper, we give conditions under which the PL approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine and thick triangulation mathcal {T}. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL approximation. Finally, we show analogous results for the PL approximation of an isomanifold with boundary.

On groups Gnk and Γnk: A study of manifolds, dynamics, and invariants

Recently, the first named author defined a 2-parametric family of groups [Formula: see text] [V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups [Formula: see text] and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of [Formula: see text] particles possess a nice codimension one property governed by exactly [Formula: see text] particles, then these dynamical systems admit a topological invariant valued in [Formula: see text]”. The [Formula: see text] groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the [Formula: see text] groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the [Formula: see text] groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance [Formula: see text] groups to get rid of this [Formula: see text]-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by [Formula: see text], which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129–145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257–1270]; the [Formula: see text] naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups [Formula: see text]: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a “braid group” of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the [Formula: see text] and [Formula: see text] theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

Discrete differential geometry and its role in computational modeling of defects and inelasticity

In this paper we discuss the geometry and mechanics of discrete manifolds—so called simplicial complexes that are essentially triangulated regions—motivated by the works on discrete exterior calculus and differential geometry in computer graphics. We show that the classical ideas related to the geometry of continuous manifolds, such as metric, curvature, and affine connections take on a much simpler and more intuitive aspect when discussed in the context of such discrete triangulated manifolds. We introduce the notion of a dual mesh to describe “dual” variables (technically differential forms). We use the dual mesh to show that the geometry of the defects such as microcracks, dislocations and incoherent boundaries as well as balance laws and constitutive relations can be introduced directly, without the need for “discretization” from a continuum. This opens up the possibility of direct simulations of these bodies without the need for a continuous counterpart. We end by demonstrating how the second law of thermodynamics can be used in such a situation including discrete non-local systems.

A combinatorial Yamabe problem on two and three dimensional manifolds

In this paper, we define a new discrete curvature on two and three dimensional triangulated manifolds, which is a modification of the well-known discrete curvature on these manifolds. The new definition is more natural and respects the scaling exactly the same way as Gauss curvature does. Moreover, the new discrete curvature can be used to approximate the Gauss curvature on surfaces. Then we study the corresponding constant curvature problem, which is called the combinatorial Yamabe problem, by the corresponding combinatorial versions of Ricci flow and Calabi flow for surfaces and Yamabe flow for 3-dimensional manifolds. The basic tools are the discrete maximal principle and variational principle.