Topological Genus Research Articles

Freestyle: Sculpting meshes with self-adaptive topology

We present a real-time method for sculpting triangular manifold meshes while enabling arbitrary surface deformation with seamless topological changes. Our insight is that the use of quasi-uniform mesh sampling, an interesting option now that very large meshes can be edited and displayed in real-time, provides the right framework for expressing and efficiently processing arbitrary changes of topological genus. The user controls deformation by gesture: he sweeps tools that apply a variety of deformation fields, from smoothing and trimming ones to local inflation and constant volume deformation tools. Meanwhile, the quasi-regular mesh seamlessly splits or locally blends when and where needed, while still following the user-specified deformation. Our method guarantees a closed, self-intersection-free mesh, whatever the user action. We demonstrate the practical usability of the resulting, interactive sculpting system through the sculpture of models that would have been extremely difficult to achieve with both current research methods and state of the art professional software.

Prediction of RNA secondary structures with pseudoknots

We present a new algorithm to predict RNA secondary structures with pseudoknots. The method is based on a classification of RNA structures according to their topological genus. The algorithm utilizes a simplified parametrization of the free energies for pair stacking, loop penalties, etc. and in addition a free energy penalty proportional to the topological genus of the pairing graph. Our method can take into account all pseudoknot topologies and achieves high success rates compared to state-of-the-art methods. This shows that the genus is a promising concept to classify pseudoknots.

Topological invariants of isolated complete intersection curve singularities

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

On q-Deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker Function I

A representation of a specialization of a q-deformed class one lattice gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed. For \ell=1, this provides an interpretation of non-specialized q-deformed gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Gamma-function as a substitute of topological genus in semi-infinite geometry. A relation with Givental-Lee universal solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.

On q-Deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker Function II

A representation of a specialization of a q-deformed class one lattice \({\mathfrak{gl}_{\ell+1}}\)-Whittaker function in terms of cohomology groups of line bundles on the space \({\mathcal{QM}_d(\mathbb{P}^{\ell})}\) of quasi-maps \({\mathbb{P}^1 \to \mathbb{P}^{\ell}}\) of degree d is proposed. For l = 1, this provides an interpretation of the non-specialized q-deformed \({\mathfrak{gl}_{2}}\)-Whittaker function in terms of \({\mathcal{QM}_d(\mathbb{P}^1)}\). In particular the (q-version of the) Mellin-Barnes representation of the \({\mathfrak{gl}_2}\)-Whittaker function is realized as a semi-infinite period map. The explicit form of the period map manifests an important role of q-version of Γ-function as a topological genus in semi-infinite geometry. A relation with the Givental-Lee universal solution (J-function) of q-deformed \({\mathfrak{gl}_2}\)-Toda chain is also discussed.

Reconstruction Algorithms as a Suitable Basis for Mesh Connectivity Compression

During a highly productive period running from 1995 to about 2002, the research in lossless compression of surface meshes mainly consisted in a hard battle for the best bitrates. However, for a few years, compression rates seem stabilized around 1.5 bit per vertex for the connectivity coding of usual triangular meshes, and more and more work is dedicated to remeshing, lossy compression, or gigantic mesh compression, where memory access and CPU optimizations are the new priority. However, the size of 3D models keeps growing, and many application fields keep requiring lossless compression. In this paper, we present a new contribution for single-rate lossless connectivity compression, which first brings improvement over current state of the art bitrates, and second, does not constraint the coding of the vertex positions, offering therefore a good complementarity with the best performing geometric compression methods. The initial observation having motivated this work is that very often, most of the connectivity part of a mesh can be automatically deduced from its geometric part using reconstruction algorithms. This has already been used within the limited framework of projectable objects (essentially, terrain models and GIS), but finds here its first generalization to arbitrary triangular meshes, without any limitation regarding the topological genus, the number of connected components, the manifoldness or the regularity. This can be obtained by constraining and guiding a Delaunay-based reconstruction algorithm so that it outputs the initial mesh to be coded. The resulting rates seem extremely competitive when the meshes are fully included in Delaunay, and are still good compared to the state-of-the-art in the case of scanned models.

“Bubble-and-grain” method and criteria for optimal positioning inhomogeneities in topological optimization

In the article we propose the enhancement of the topology optimization method, which uses an iterative positioning, orientation and hierarchical shape optimization of subsequently introduced elastic inhomogeneities. The inserted elastic inhomogeneities could be more or less compliant than the elastic medium of the structural element being optimized. One extreme case of the inhomogeneity is the cavity of zero stiffness (“bubble”), while the other limit corresponds to the absolutely rigid inhomogeneity (“grain”). This extension of the topology method requires the generalization of topological derivatives. The topological derivative is an instrument for solving topology optimization problems. Namely, the topological derivative quantifies the sensitivity of a problem when the domain under consideration is perturbed by changing its topological genus. In this article we represent the generalized topological derivatives exploiting the Eshelby approach of effective inhomogeneity. For this purpose we study sensitivity of the optimization functional to the placement of infinitesimally small elliptical inhomogeneity. The sensitivity to the infinitesimal translation of inclusion is quantified by the characteristic function. The infinitesimally small inhomogeneity must be inserted at the point, where the characteristic function attains its extreme value. Next, we examine the sensitivity of the Lagrangian to orientation of ellipse and determine its optimal orientation. Finally, we express the optimal eccentricity of ellipse as the function of averaged principal strains in inhomogeneous medium. The compliance functional plays the role of optimization criterion. Using adjoint variables technique of variational calculus, the results could be extended for arbitrary integral functionals.

On the connections between the topological genus of certain polyhedra and the algebraic genus of their Hilton-Hopf quadratic forms

The Hilton-Hopf quadratic form is defined for spaces of the homotopy type of a CW complex with one cell each in dimensions 0 and 4n, K cells in dimension 2n and no other cells. If two such spaces are of the same topological genus, then their Hilton-Hopf quadratic forms are of the same weak algebraic genus. For large classes of spaces, such as simply connected differentiable 4-manifolds, the converse is also true, as long as the suspensions of the spaces are also of the same topological genus. This note allays the conjecture that the converse is true in general by offering two techniques for generating infinite families of counterexamples.

Topology of non-linear structure in the 2dF Galaxy Redshift Survey

We study the evolution of non-linear structure as a function of scale in samples from the 2dF Galaxy Redshift Survey, constituting over 221 000 galaxies at a median redshift of z=0.11. The two flux-limited galaxy samples, located near the southern galactic pole and the galactic equator, are smoothed with Gaussian filters of width ranging from 5 to 8 Mpc/h to produce a continuous galaxy density field. The topological genus statistic is used to measure the relative abundance of overdense clusters to void regions at each scale; these results are compared to the predictions of analytic theory, in the form of the genus statistic for i) the linear regime case of a Gaussian random field; and ii) a first-order perturbative expansion of the weakly non-linear evolved field. The measurements demonstrate a statistically significant detection of an asymmetry in the genus statistic between regions corresponding to low- and high-density volumes of the universe. We attribute the asymmetry to the non-linear effects of gravitational evolution and biased galaxy formation, and demonstrate that these effects evolve as a function of scale. We find that neither analytic prescription satisfactorily reproduces the measurements, though the weakly non-linear theory yields substantially better results in some cases, and we discuss the potential explanations for this result.

Automorphism groups of q-trigonal planar Klein surfaces and maximal surfaces

A compact Klein surface X=D/Γ , where D denotes the hyperbolic plane and Γ is a surface NEC group, is said to be q-trigonal if it admits an automorphism ϕ of order 3 such that the quotient X/<ϕ > has algebraic genus q. In this paper we obtain for each q the automorphism groups of q-trigonal planar Klein surfaces, that is surfaces of topological genus 0 with k≥ 3 boundary components. We also study the surfaces in this family, which have an automorphism group of maximal order (maximal surfaces). It will be done from an algebraic and geometrical point of view.

Quadrilateral mesh simplification

We introduce a simplification algorithm for meshes composed of quadrilateral elements. It is reminiscent of edge-collapse based methods for triangle meshes, but takes a novel approach to the challenging problem of maintaining the quadrilateral connectivity during level-of-detail creation. The method consists of a set of unit operations applied to the dual of the mesh, each designed to improve mesh structure and maintain topological genus. Geometric shape is maintained by an extension of a quadric error metric to quad meshes. The technique is straightforward to implement and efficient enough to be applied to real-world models. Our technique can handle models with sharp features, and can be used to re-mesh general polygonal, i.e. tri- and quad-dominant, meshes into quadonly meshes.

Topological Classification of RNA Structures

We present a novel topological classification of RNA secondary structures with pseudoknots. It is based on the topological genus of the circular diagram associated to the RNA base-pair structure. The genus is a positive integer number whose value quantifies the topological complexity of the folded RNA structure. In such a representation, planar diagrams correspond to pure RNA secondary structures and have zero genus, whereas non-planar diagrams correspond to pseudoknotted structures and have higher genus. The topological genus allows for the definition of topological folding motifs, similar in spirit to those introduced and commonly used in protein folding. We analyze real RNA structures from the databases Worldwide Protein Data Bank and Pseudobase and classify them according to their topological genus. For simplicity, we limit our analysis by considering only Watson–Crick complementary base pairs and G–U wobble base pairs. We compare the results of our statistical survey with existing theoretical and numerical models. We also discuss possible applications of this classification and show how it can be used for identifying new RNA structural motifs.

Robust parametrization of brain surface meshes

Complex shapes - such as the surface of the human brain - may be represented and analyzed in frequency space by means of a spherical harmonics transformation. A key step of the processing chain is introducing a suitable parametrization of the triangular mesh representing the brain surface. This problem corresponds to mapping a surface of topological genus zero on a unit sphere. An algorithm is described that produces an optimal combination of an area- and angle-preserving mapping. A multi-resolution scheme provides the robustness required to map the highly detailed and convoluted brain surface. More than 1000 datasets were successfully processed by this mature and robust approach.

Theory of genus reduction in alkali-induced graphitization of nanoporous carbon

Exposure to elemental Cs generates graphitic domains within nanoporous carbon at only $50\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, well below the typical graphitization temperatures of $&gt;1000\phantom{\rule{0.2em}{0ex}}\ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$. We present a model of nanoporous carbon, the wormhole, which can express the fundamental topological elements of graphitization: a negative-curvature analog to ${\mathrm{C}}_{60}$ fullerene. The pathway to wormhole annihilation comprises an initial Stone-Wales transformation and a subsequent unzipping of the defect. This complex $\ensuremath{\sim}100$-atom collective defect disintegrates with the formation of only two dangling bonds. Our ab initio calculations show that while the activation barrier against reduction in topological genus is indeed lowered by interaction with alkali, an additional chemical constituent must be involved to account for this remarkable local graphitization of nanoporous carbons at near-room temperature.

Equilibrium shape degeneracy in starfish vesicles

Phospholipid dispersed in aqueous buffer above their critical micellar concentration form bilayers, which can spontaneously adopt closed metastable shapes with sizes ranging from a few hundred nanometers to few tens of micrometers. The equilibrium shapes of these vesicles are well described by the Canham-Evans-Helfrich curvature elastic energy. Their floppiness allows for thermal fluctuations to be easily detected, but no spontaneous shape transformation is usually observed for vesicles of spherical topological genus (i.e., shapes with no holes) because of strict geometrical constraints and/or energy barriers. This report shows that for a particular class of shapes with spherical topological genus (starfish vesicles), dramatic spontaneous shape transformations can occur due to the degeneracy of the shape solutions, as demonstrated by numerical calculations. These predictions are supported by experimental observations of a three-arm starfish vesicle undergoing spontaneous shape transformations similar to those predicted numerically.

On q-n-gonal Klein Surfaces

We consider proper Klein surfaces X of algebraic genus p ≥ 2, having an automorphism ϕ of prime order n with quotient space X/〈ϕ〉 of algebraic genus q. These Klein surfaces are called q-n-gonal surfaces and they are n-sheeted covers of surfaces of algebraic genus q. In this paper we extend the results of the already studied cases n ≤ 3 to this more general situation. Given p ≥ 2, we obtain, for each prime n, the (admissible) values q for which there exists a q-n-gonal surface of algebraic genus p. Furthermore, for each p and for each admissible q, it is possible to check all topological types of q-n-gonal surfaces with algebraic genus p. Several examples are given: q-pentagonal surfaces and q-n-gonal bordered surfaces with topological genus g = 0, 1.

Topology of pseudoknotted homopolymers

We consider the folding of a self-avoiding homopolymer on a lattice, with saturating hydrogen bond interactions. Our goal is to numerically evaluate the statistical distribution of the topological genus of pseudoknotted configurations. The genus has been recently proposed for classifying pseudoknots (and their topological complexity) in the context of RNA folding. We compare our results on the distribution of the genus of pseudoknots, with the theoretical predictions of an existing combinatorial model for an infinitely flexible and stretchable homopolymer. We thus obtain that steric and geometric constraints considerably limit the topological complexity of pseudoknotted configurations, as it occurs for instance in real RNA molecules. We also analyze the scaling properties at large homopolymer length, and the genus distributions above and below the critical temperature between the swollen phase and the compact-globule phase, both in two and three dimensions.

Topological instability along invariant surfaces and pseudochaotic transport

The paper describes the complex topological structure of invariant surfaces that appears in a quasi-stationary regime of the tokamak plasma, and it considers in detail anomalous transport of particles along the invariant surfaces (isosurfaces) that have topological genus greater than 1. Such dynamics is pseudochaotic; i.e. it has a zero Lyapunov exponent. Simulations discover such surfaces in confined plasmas under a fairly low ratio of pressure to the magnetic field energy (beta). The isosurfaces correspond to quasi-coherent structures called "streamers" and the streamers are connected by filaments. We study distribution of time of particle separation, Poincaré; recurrences of trajectories, and first time arrival to the system's edge. A model of a multibar-in-square billiard, introduced by Carreras et al. [Chaos 13, 1175 (2003)] is studied with renormalization group method to obtain a distribution of the first time of particles arrival to the edge as a function of the number of bars, which appears to be power-like. The characteristic exponent of this distribution is discussed with respect to its dependence on the number of filaments that connect adjacent streamers.

On matrix models for anomalous dimensions of super-Yang–Mills theory

We consider the matrix model approach to the anomalous dimension matrix in N = 4 super-Yang–Mills theory. We construct the path integral representation for the anomalous dimension density matrix and analyze the resulting action. In particular, we consider the large N limit, which results in a classical field theory. Since the same limit leads to spin chains, we propose to consider the former as an alternative description of the latter. We consider also the limit of small N, which corresponds to the restriction to the diagrams of maximal topological genus.

Skeleton-driven 2D distance field metamorphosis using intrinsic shape parameters

In this article a novel algorithm is presented for 2D shape interpolation using the intrinsic shape parameters of a piecewise linear curve. The skeletons of two given shapes are computed and the smooth transformation of distance fields is driven by metamorphosis from the skeleton of the source object to that of the target one. We introduce feature graphs, linear forms of skeletons, to guide the construction of intermediate skeleton. If the topologies of the source object and the target one are different, their feature graphs will be automatically extended with equivalent topologies. Then we apply the technique of intrinsic shape parameters to the smooth transition of the extended feature graphs, which will guide the metamorphosis of the skeletons. Not only can the new approach be capable of morphing between objects with different topological genus, but also the topologies and the shapes of the intermediate objects can be controlled efficiently.