Triangulations Of Manifolds Research Articles

Euclidean 4-Simplices and Invariants of Four-Dimensional Manifolds: III. Moves 1 ↔ 5 and Related Structures

We conclude the construction of the algebraic complex, consisting of spaces of differentials of Euclidean metric values, for four-dimensional piecewise-linear manifolds. Assuming that the complex is acyclic, we investigate how its torsion changes under rebuildings of the manifold triangulation. First, we write out formulas for moves 3 -> 3 and 2 <-> 4 based on the results of our two previous works, and then we study in detail moves 1 <-> 5. On this basis, we obtain the formula for a four-dimensional manifold invariant. As an example, we present a detailed calculation of our invariant for sphere S^4; in particular, the complex turns out, indeed, to be acyclic.

Distinguishing Three-Dimensional Lens Spaces L(7,1) and L(7,2) by Means of Classical Pentagon Equation

We construct new topological invariants of three-dimensional manifolds which can, in particular, distinguish homotopy equivalent lens spaces L(7, 1) and L(7, 2). The invariants are built on the base of a classical (not quantum) solution of pentagon equation, i.e. algebraic relation corresponding to a “2 tetrahedra → 3 tetrahedra” local re-building of a manifold triangulation. This solution, found earlier by one of the authors, is expressed in terms of metric characteristics of Euclidean tetrahedra.

Formula omitted] modeling with A-patches from rational trivariate functions

We approximate a manifold triangulation in R 3 using smooth implicit algebraic surface patches, which we call A-patches. Here each A-patch is a real iso-contour of a trivariate rational function defined within a tetrahedron. The rational trivariate function provides increased degrees of freedom so that the number of surface patches needed for free-form shape modeling is significantly reduced compared to earlier similar approaches. Furthermore, the surface patches have quadratic precision, that is they exactly recover quadratic surfaces. We give conditions under which a C 1 smooth and single sheeted surface patch is isolated from the multiple sheets.

The Gromov norm and foliations

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology. We show that this norm is non-trivial — i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold.¶Using a homotopy-theoretic refinement, we show that a taut foliation whose leaf space branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut foliation whose leaf space branches in both directions. Our technology also lets us produce examples of taut foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover.¶Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties.

Foliations transverse to triangulations of 3-manifolds

We investigate the combinatorial analogues, in the context of normal surfaces, of taut and transversely measured (codimension 1) foliations of 3-manifolds. We establish that the existence of certain combinatorial structures, a priori weaker than the existence of the corresponding foliation, is sufficient to guarantee that the manifold in question satisfies certain properties, e.g. irreducibility. The finiteness of our combinatorial structures allows us to make our results quantitative in nature and has (coarse) geometrical consequences for the manifold. Furthermore, our techniques give a straightforward combinatorial proof of Novikov's theorem.

A TOPOLOGICAL INVARIANT OF RG FLOWS IN 2D INTEGRABLE QUANTUM FIELD THEORIES

We construct a topological invariant of the renormalization group trajectories of a large class of 2D quantum integrable models, described by the thermodynamic Bethe ansatz approach. A geometrical description of this invariant in terms of triangulations of three-dimensional manifolds is proposed and associated dilogarithm identities are proven.

The average edge order of triangulations of 3-manifolds with boundary

Feng Luo and Richard Stong introduced the average edge order $\mu _0(K)$ of a triangulation $K$ and showed in particular that for closed 3-manifolds $\mu _0(K)$ being less than 4.5 implies that $K$ is on $S^3$. In this paper, we establish similar results for 3-manifolds with non-empty boundary; in particular it is shown that $\mu _0(K)$ being less than 4 implies that $K$ is on the 3-ball.

Linear Conditions on the Number of Faces of Manifolds with Boundary

The Euler equation and the Dehn–Sommerville equations are known to be the only (rational) linear conditions forf-vectors (number of simplices at various dimensions) of triangulations of spheres. We generalize this fact to arbitrary triangulations, linear triangulations of manifolds, and polytopal triangulations of Euclidean balls. We prove that for closed manifolds, the Euler equation and the Dehn–Sommerville equations remain the only linear conditions. We also prove that for manifolds with nonempty boundary, the Euler equation is the only linear condition. These results are proved not only over Q, but also over Z and Z/kZ.

Chaotic actions of free groups

An action of a group, G, on a space, M, is chaotic if it is topologically transitive and the set of points with finite orbit is a dense subset of M. In this paper we show that every compact triangulable manifold of dimension greater than one admits a faithful chaotic action of every countably generated free group.

The average edge order of triangulations of 3-manifolds

The average edge order of triangulations of 3-manifolds

Flip-moves and graded associative algebras

The relation between discrete topological field theories on triangulations of two-dimensional manifolds and associative algebras was worked out recently. The starting point for this development was the graphical interpretation of the associativity as flip of triangles. We show that there is a more general relation between flip-moves with two $n$-gons and $Z_{n-2}$-graded associative algebras. A detailed examination shows that flip-invariant models on a lattice of $n$-gons can be constructed {}from $Z_{2}$- or $Z_{1}$-graded algebras, reducing in the second case to triangulations of the two-dimensional manifolds. Related problems occure naturally in three-dimensional topological lattice theories.

SMART: a solvent-accessible triangulated surface generator for molecular graphics and boundary element applications.

An algorithm is presented for generating a representation of the solvent-accessible molecular surface as a smooth triangulated manifold. The algorithm, called SMART (SMooth moleculAR surface Triangulator), divides the contact and reentrant portions of the solvent-accessible molecular surface into curvilinear three-sided elements. In contrast to the author's earlier implementation of this general approach [Zauhar, R.J. and Morgan, R.S., J. Comput. Chem., 11 (1990) 603], the SMART algorithm defines elements directly on the appropriate geometric surface types (rather than using interpolation over cubic elements), and has special features to handle highly distorted regions which often appear in deep crevices and internal cavities. While the method is designed for use with boundary element techniques in continuum electrostatics, it can also be applied to the accurate computation of molecular surface areas and volumes, and the generation of shaded surfaces for display with interactive computer graphics.

Invariants from triangulations of hyperbolic 3-manifolds

For any finite volume hyperbolic 3-manifold M we use ideal triangulation to define an invariant β(M) in the Bloch group B(C ). It actually lies in the subgroup of B(C ) determined by the invariant trace field of M . The Chern-Simons invariant of M is determined modulo rationals by β(M). This implies rationality and — assuming the Ramakrishnan conjecture — irrationality results for Chern Simons invariants.

Complementary colorings

Two 3-colorings of a cycle are complementary if whenever a vertex has its neighbors colored alike in one coloring, they are colored differently in the other coloring. Describing complementary colorings in terms of heawood colorings, we are able to count all such pairs. Complementary colorings can be defined for triangulations of manifolds. We construct complementary colorings for all oriented surfaces and for the 3-sphere. Finally, we apply these colorings to constructing triangulations whose odd part is a manifold.

The Average Edge Order of 3-Manifold Coloured Triangulations

AbstractIf K is a triangulation of a closed 3-manifold M with E0(K) edges and F0(K) triangles, then the average edge order of K is defined to beIn [8], the relations between this quantity and the topology of M are investigated, especially in the case of μ0(K) being small (where the study relies on Oda's classification of triangulations of 𝕊2 up to eight vertices—see [9]). In the present paper, the attention is fixed upon the average edge order of coloured triangulations; surprisingly enough, the obtained results are perfectly analogous to Luo-Stong' ones, and may be proved with little effort by means of edge-coloured graphs representing manifolds.

Combinatorics of Triangulations of 3-Manifolds

Combinatorics of Triangulations of 3-Manifolds

Combinatorics of triangulations of 3-manifolds

In this paper, we study the average edge order of triangulations of closed 3 3 -manifolds and show in particular that the average edge order being less than 4.5 4.5 implies that triangulation is on the 3 3 -sphere.

Minimum ideal triangulations of hyperbolic 3-manifolds

Let ?(n) be the minimum number of ideal hyperbolic tetrahedra necessary to construct a finite volumen-cusped hyperbolic 3-manifold, orientable or not. Let ?or(n) be the corresponding number when we restrict ourselves to orientable manifolds. The correct values of ?(n) and ?or(n) and the corresponding manifolds are given forn=1,2,3,4 and 5. We then show that 2n?1≤?(n)≤?or(n)≤4n?4 forn?5 and that ?or(n)?2n for alln.

The combinatorics of colored triangulations of manifolds

Foundations for the topic of crystallizations are proposed through the more general concept of colored triangulations. Classic results and techniques of crystallizations are reviewed from this point of view. A new set of combinatorial invariants of manifolds is defined, and related to the fundamental group and other known invariants. A universal group theoretic approach for this theory is introduced.

Fixed point sets of deformations of pairs of spaces

Let ( X, A) be a pair of compact polyhedra which satisfies conditions similar to those needed in order to realize the Nielsen number of the identity map in the cases where A = ø or A = X. Deformations of ( X, A) with minimal fixed point sets are constructed, and are used to construct deformations of ( X, A) with prescribed fixed point sets. The size and location of these fixed point sets depends only on the Euler characteristics of the components of X and A. As an application it follows that if M is an even-dimensional compact connected triangulable manifold with boundary Bd M, then there is no difference between possible fixed point sets of deformations of M and of deformations of ( M, Bd M).