Nonzero Genus Research Articles

Curves are algebraic K(π,1)$K(\pi,1)$: Theoretical and practical aspects

AbstractWe prove that any geometrically connected curve over a field is an algebraic , as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of ‐modules, with invertible in , is canonically isomorphic to the cohomology of its corresponding ‐module. To this end, we explicitly construct some Galois coverings of corresponding to Galois coverings of the normalisation of its irreducible components. When is finite or algebraically closed, we precisely describe finite quotients of that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products and in étale cohomology in terms of finite group cohomology.

Geodesic conformal gradient device based on a torus.

In recent years, optical fields on non-Euclidean geometry have become a hot topic. The geodesic conformal transformation theory, linking curved surfaces with planar gradient refractive indices, holds unique advantages in controlling curved optical fields. However, this theory has not yet addressed surfaces with non-trivial topology with a certain genus. In this work, we design a gradient planar device based on the geodesic conformal transformation theory for toroidal surfaces, which can achieve Gaussian beam focusing. Unlike traditional angle-preserving geodesic theories, the non-zero genus results in the one-to-two discontinuous boundaries in the device, and we utilize inversion transformations to rectify this drawback.

Fluid Cohomology

The vorticity-streamfunction formulation for incompressible inviscid fluids is the basis for many fluid simulation methods in computer graphics, including vortex methods, streamfunction solvers, spectral methods, and Monte Carlo methods. We point out that current setups in the vorticity-streamfunction formulation are insufficient at simulating fluids on general non-simply-connected domains. This issue is critical in practice, as obstacles, periodic boundaries, and nonzero genus can all make the fluid domain multiply connected. These scenarios introduce nontrivial cohomology components to the flow in the form of harmonic fields. The dynamics of these harmonic fields have been previously overlooked. In this paper, we derive the missing equations of motion for the fluid cohomology components. We elucidate the physical laws associated with the new equations, and show their importance in reproducing physically correct behaviors of fluid flows on domains with general topology.

Part-to-Surface Mesh Segmentation for Mechanical Models Based on Multi-Stage Clustering

In this paper, we propose a part-to-surface mesh segmentation method for engineering models characterized by a multi-stage clustering of surface patches generated by the Variational Shape Approximation (VSA) method. Initially, border types between topologically adjacent surface patches are estimated and surface patches bonded by certain border types are grouped into clusters. Secondly, clusters of surface patches are iteratively aggregated into volumetric part-level sub-meshes according to the convexity variations of intermediate sub-meshes. At last, surface patches within their respective part-level sub-meshes are merged into surface-level sub-meshes that are well-fitted by basic primitive shapes based on an improved surface-aware similarity metric. Experimental results demonstrate that the proposed method can achieve better results on mechanical models both quantitatively and perceptually, especially for objects with non-zero genus topology and rich in tiny features.

The shape distribution of superclusters in SDSS DR12

ABSTRACTGalaxy superclusters, the largest galaxy structures in the cosmic web, are formed due to the gravitational collapse (although they are not usually gravitationally bound). Their geometrical properties can shed light on the structure formation process on cosmological scales, hence on the fundamental properties of gravity itself. In this work we study the distributions of the shape, topology, and morphology of the superclusters extracted from Sloan Digital Sky Survey Data Release 12 (SDSS DR12) main galaxy sample and defined in two different ways – using fixed and adaptive density threshold in the luminosity-density field. To assess the geometry and topology of each individual supercluster, we employ Minkowski functionals and Shapefinders, precisely calculated by the shape diagnostic tool SURFGEN2. Both supercluster samples produce similar shape distributions. Perhaps not surprisingly, most superclusters are spherical in shape with trivial topology. However, large superclusters with volumes V ≳ 104 Mpc3 are statistically found to be filamentary with non-zero genus values. The results, supercluster catalogues and shape distributions are publicly available.

Volume decomposition for multi-axis support-free and gouging-free printing based on ellipsoidal slicing

Continuous multi-axis printing has drawn more and more attention owing to its unique advantages of eliminating the need of support (at least in theory) when printing complex models with overhang features and significantly reducing the stair-step effect of the conventional three-axis printing. For continuous multi-axis printing, the printing layers are typically not planar but curved, so to capitalize the extra DOFs provided by the rotary axes of the printer. When a printing layer is concave (i.e., its curvature vector points inward at the printing point), the potential nozzle-layer local gouging becomes a serious concern, which is especially problematic for metal printing wherein the size of the nozzle head is usually large. Unfortunately, this convexity issue has received little attention in the existing curved slicing algorithms, making them prone to the local gouging quandary. In this paper, we propose a novel curved layer decomposition algorithm based on the use of (only the positive sides of) ellipsoidal surfaces. To facilitate the generation of ellipsoidal printing layers for an arbitrary freeform solid model with a complex topology (e.g., having a non-zero genus number), the powerful algebraic technique of skeletonization is employed by our algorithm to first decompose the model into a suitable set of sub-entities, for each of which a series of ellipsoidal printing layers are then generated, wherein the parameters of ellipsoidal layers are adaptively adjusted to satisfy the necessary constraints such as support-free and other requirements. Preliminary experiments in both computer simulation and physical printing of the proposed algorithm are conducted, and the results give a positive validation on the effectiveness and feasibility of the proposed methodology.

Adaptive anisotropic porous structure design and modeling for 2.5D mechanical parts

As displaying many advantages including high specific strength, light weight, energy absorption, etc., porous structures have been widely used in aerospace, medical science, engineering and other fields. In this paper, an adaptive anisotropic porous modeling method is proposed for improving the mechanical performance of 2.5D mechanical parts whilst minimizing weight and maximizing product benefits. The method relies on the Anisotropic Centroidal Voronoi Tessellations (ACVTs) by offsetting closed B-spline curves. To enhance adaptability to a given parts, the finite element analysis (FEA) results of stress field is combined with stress-based weighted random sampling strategy to generate ACVTs according to Riemannian metric. A skin frame is established to improve geometrical quality of porous structure and ACVTs is tailored to be adaptive for concave and Non-zero genus. Besides, a parametric model between the stress tensor distribution and the relative density field is formulated, allowing the size, the distribution and the shape of pores to be controlled by stress mapping. Both the FEA results and the experimental data show that the adaptive anisotropic structures have significant advantages in mechanical properties, topological consistency and connectivity. Additionally, an example of wrench is provided to prove the great potential of the adaptive anisotropic structures in engineering applications.

DNA origami and unknotted A-trails in torus graphs

Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated embeddings. Surface meshes are frequent targets for DNA nanostructure self-assembly, and so we study both triangular and rectangular torus grids. We show that aside from one exceptional family, a triangular torus grid contains an A-trail if and only if it has an odd number of vertices, and that such an A-trail is necessarily unknotted. On the other hand, while every rectangular torus grid contains an unknotted A-trail, we also show that any torus knot can be realized as an A-trail in some rectangular grid. Lastly, we use a gluing operation to construct infinite families of triangular and rectangular grids containing unknotted A-trails on surfaces of arbitrary genus. We also give infinite families of triangular grids containing no unknotted A-trail on surfaces of arbitrary nonzero genus.

Minimal-Area Metrics on the Swiss Cross and Punctured Torus

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2\pi. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the "Swiss cross" challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.

On noncollapsed almost Ricci-flat 4-manifolds

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the fundamental group is infinite or, more generally, of sufficiently large cardinality.

More on homological supersymmetric quantum mechanics

In this work, we first solve complex Morse flow equations for the simplest case of a bosonic harmonic oscillator to discuss localization in the context of Picard-Lefschetz theory. We briefly touch on the exact non-BPS solutions of the bosonized supersymmetric quantum mechanics on algebraic geometric grounds and report that their complex phases can be accessed through the cohomology of WKB 1-form of the underlying singular spectral curve subject to necessary cohomological corrections for non-zero genus. Motivated by Picard-Lefschetz theory, we write down a general formula for the index of $\mathcal{N} = 4$ quantum mechanics with background $R$-symmetry gauge fields. We conjecture that certain symmetries of the refined Witten index and singularities of the moduli space may be used to determine the correct intersection coefficients. A few examples, where this conjecture holds, are shown in both linear and closed quivers with rank-one quiver gauge groups. The $R$-anomaly removal along the "Morsified" relative homology cycles also called "Lefschetz thimbles" is shown to lead to the appearance of Stokes lines. We show that the Fayet-Iliopoulos (FI) parameters appear in the intersection coefficients for the relative homology of the quiver quantum mechanics resulting from dimensional reduction of $2d$ $\mathcal{N}=(2,2)$ gauge theory on a circle and explicitly calculate integrals along the Lefschetz thimbles in $\mathcal{N}=4$ $\mathbb{CP}^{k-1}$ model. The Stokes jumping of coefficients and its relation to wall crossing phenomena is briefly discussed. We also find that the notion of "on-the-wall" index is related to the invariant Lefschetz thimbles under Stokes phenomena. An implication of the Lefschetz thimbles in constructing knots from quiver quantum mechanics is indicated.

Fundamental domains and generators for lattice Veech groups

The moduli space $Q \mathcal M\_g$ of non-zero genus $g$ quadratic differentials has a natural action of $G=\mathrm {GL}\_2^+(\mathbb R)$ / $\langle ± \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$ $\rangle$. The Veech group PSL$(X,q)$ is the stabilizer of $(X,q) \in Q \mathcal M\_g$ in $G$. We describe a new algorithm for finding elements of PSL$(X,q)$ which, for lattice Veech groups, can be used to compute a fundamental domain and generators. Using our algorithm, we give the first explicit examples of generators and fundamental domains for non-arithmetic Veech groups where the genus of $\mathbb H$ / PSL$(X,q)$ is greater than zero.

Maps, immersions and permutations

We consider the problem of counting and of listing topologically inequivalent “planar” 4-valent maps with a single component and a given number [Formula: see text] of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options, where the circle and/or the sphere are/is oriented are considered in turn, following Arnold’s classification of the different types of symmetries. We also consider the case of bicolorable and bicolored maps or immersions, where faces are bicolored. Our method extends to immersions of a circle in a higher genus Riemann surface. There the bicolorability is no longer automatic and has to be assumed. We thus have two separate countings in nonzero genus, that of bicolorable maps and that of general maps. We use a classical method of encoding maps in terms of permutations, on which the constraints of “one-componentness” and of a given genus may be applied. Depending on the orientation issue and on the bicolorability assumption, permutations for a map with [Formula: see text] vertices live in [Formula: see text] or in [Formula: see text]. In a nutshell, our method reduces to the counting (or listing) of orbits of certain subset of [Formula: see text] (respectively, [Formula: see text]), under the action of the centralizer of a certain element of [Formula: see text] (respectively, [Formula: see text]). This is achieved either by appealing to a formula by Frobenius or by a direct enumeration of these orbits. Applications to knot theory are briefly mentioned.

Smooth tool path generation for 5-axis machining of triangular mesh surface with nonzero genus

NC machining of a nonzero genus triangular mesh surface is being more widely confronted than before in the manufacturing field. At present, due to the complexity of geometry computation related to tool path generation, only one path pattern of iso-planar type is adopted in real machining of such surface. To improve significantly 5-axis machining of the nonzero genus mesh surface, it is necessary to develop a more efficient and robust tool path generation method. In this paper, a new method of generating spiral or contour-parallel tool path is proposed, which is inspired by the cylindrical helix or circle which are a set of parallel lines on the rectangular region obtained by unwrapping the cylinder. According to this idea, the effective data structure and algorithm are first designed to transform a nonzero genus surface into a genus-0 surface such that the conformal map method can be used to build the bidirectional mapping between the genus-0 surface and the rectangular region. In this rectangular region, the issues of spiral or contour-parallel tool path generation fall into the category of simple straight path planning. Accordingly, the formula for calculating the parameter increment for the guide line is derived by the difference scheme on the mesh surface and an accuracy improvement method is proposed based on the edge curve interpolation for determining the cutter contact (CC) point. These guarantee that the generated tool path can meet nicely the machining requirement. To improve further the kinematic and dynamic performance of 5-axis machine tool, a method for optimizing tool orientation is also preliminarily investigated. Finally, the experiments are performed to demonstrate the proposed method and show that it can generate nicely the spiral tool path or contour-parallel tool path on the nonzero genus mesh surface and also can guarantee the smooth change of tool orientation.

Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian

We consider cohomological and Poisson structures associated with the special tautological subbundles \(TB_{W_{1,2, \ldots n} }\) for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of \(TB_{W_{1,2, \ldots n} }\) are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev-Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in \(TB_{W_{1,2, \ldots n} }\) can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in \(TB_{W_{\hat S} }\) and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.

A riemann-hilbert problem in a riemann surface

One of the inspirations behind Peter Lax's interest in dispersive integrable systems, as the small dispersion parameter goes to zero, comes from systems of ODEs discretizing 1-dimensional compressible gas dynamics [17]. For example, an understanding of the asymptotic behavior of the Toda lattice in different regimes has been able to shed light on some of von Neumann's conjectures concerning the validity of the approximation of PDEs by dispersive systems of ODEs.Back in the 1990s several authors have worked on the long time asymptotics of the Toda lattice [2, 7, 8, 19]. Initially the method used was the method of Lax and Levermore [16], reducing the asymptotic problem to the solution of a minimization problem with constraints (an “equilibrium measure” problem). Later, it was found that the asymptotic method of Deift and Zhou (analysis of the associated Riemann-Hilbert factorization problem in the complex plane) could apply to previously intractable problems and also produce more detailed information.Recently, together with Gerald Teschl, we have revisited the Toda lattice; instead of solutions in a constant or steplike constant background that were considered in the 1990s we have been able to study solutions in a periodic background.Two features are worth noting here. First, the associated Riemann-Hilbert factorization problem naturally lies in a hyperelliptic Riemann surface. We thus generalize the Deift-Zhou “nonlinear stationary phase method” to surfaces of nonzero genus. Second, we illustrate the important fact that very often even when applying the powerful Riemann-Hilbert method, a Lax-Levermore problem is still underlying and understanding it is crucial in the analysis and the proofs of the Deift-Zhou method!

Kutasov-like duality from D5-branes wrapping hyperbolic cycles

We study the addition of N f flavor D5-branes to supergravity solutions describing D5-branes wrapping two-cycles of genus g > 1 inside a six-dimensional space equipped with an SU ( 3 ) -structure. The non-zero genus g on the gravity side is dual to the existence of massless adjoint chiral superfields. Three types of internal manifolds are considered, each involving one of the following fibered products: H 2 × SL ˜ 2 , S 2 × SL ˜ 2 or H 2 × S 3 , where SL ˜ 2 stands for the universal cover of SL ( 2 , R ) . For the first one, we investigate the dual field theories. We show that some of the solutions with N f ≠ 0 are dual to four-dimensional N = 1 field theories exhibiting a Kutasov-like duality taking N c → k N f − N c and keeping N f fixed. Computed from the supergravity picture, k is in general a rational number, which can be made integer to fit the expectation from the field theory side. We finally study some other properties of those field theories.

Iterative two-step genetic-algorithm-based method for efficient polynomial B-spline surface reconstruction

Surface reconstruction is a very challenging problem arising in a wide variety of applications such as CAD design, data visualization, virtual reality, medical imaging, computer animation, reverse engineering and so on. Given partial information about an unknown surface, its goal is to construct, to the extent possible, a compact representation of the surface model. In most cases, available information about the surface consists of a dense set of (either organized or scattered) 3D data points obtained by using scanner devices, a today’s prevalent technology in many reverse engineering applications. In such a case, surface reconstruction consists of two main stages: (1) surface parameterization and (2) surface fitting. Both tasks are critical in order to recover surface geometry and topology and to obtain a proper fitting to data points. They are also pretty troublesome, leading to a high-dimensional nonlinear optimization problem. In this context, present paper introduces a new method for surface reconstruction from clouds of noisy 3D data points. Our method applies the genetic algorithm paradigm iteratively to fit a given cloud of data points by using strictly polynomial B-spline surfaces. Genetic algorithms are applied in two steps: the first one determines the parametric values of data points; the later computes surface knot vectors. Then, the fitting surface is calculated by least-squares through either SVD (singular value decomposition) or LU methods. The method yields very accurate results even for surfaces with singularities, concavities, complicated shapes or nonzero genus. Six examples including open, semi-closed and closed surfaces with singular points illustrate the good performance of our approach. Our experiments show that our proposal outperforms all previous approaches in terms of accuracy and flexibility.

Birational cobordism invariance of uniruled symplectic manifolds

A symplectic manifold $(M,\omega)$ is called {\em (symplectically) uniruled} if there is a nonzero genus zero GW invariant involving a point constraint. We prove that symplectic uniruledness is invariant under symplectic blow-up and blow-down. This theorem follows from a general Relative/Absolute correspondence for a symplectic manifold together with a symplectic submanifold. A direct consequence is that symplectic uniruledness is a symplectic birational invariant. Here we use Guillemin and Sternberg's notion of cobordism as the symplectic analogue of the birational equivalence.

Toroidal polyhexes, tripartite graphs, and double groups

The concept of regular Platonic polyhedra (namely the tetrahedron, octahedron, icosahedron, and their duals) can be generalized to regular polygonal networks on surfaces of non-zero genus known as Platonic tessellations. Such tessellations are useful in molecular physics as geometrical models for double groups as well as for structures of low-density carbon and boron nitride allotropes. The symmetries of such tessellations are described by automorphism groups, which generally are non-Euclidean groups going sometimes far beyond the standard symmetry point groups. A special type of Platonic tessellation, namely the duals of the regular tripartite graphs K n , n , n , can either be depicted as a tessellation of regular 2n-gons on a surface of genus (n − 2)(n − 1)/2 or as an n × n rhomboidal array of n 2 regular hexagons on a torus, called a toroidal polyhex. The simplest such tessellation is the cube considered as the dual of the octahedron (the K 2,2,2 graph), which has a point group symmetry O h with a rotational subgroup O isomorphic to the tetrahedral group T d having operations of periods 2, 3, and 4. The 2 × 2 toroidal polyhex corresponding to the cube likewise has O h as its automorphism group, but its proper rotation subgroup is isomorphic to the tetrahedral group T h having operations of periods 2, 3 and 6. A related more complicated case, recently studied by Ceulemans, Lijnens, and Fowler, is the Dyck tessellation of 12 octagons on a genus 3 surface of interest as a model for a possible carbon allotrope containing six- and eight-membered rings. The Dyck tessellation is the dual of the K 4,4,4 graph embedded in a genus 3 surface and corresponds to a 4 × 4 toroidal polyhex. Related p × q (p ≠ q) toroidal polyhex embeddings also appear to be at least feasible for Platonic tessellations corresponding to double and higher multiple groups of certain Platonic polyhedra and other tessellations. Thus the 2 × 4 toroidal polyhex is seen to correspond to the genus 2 {4 + 4, 3} tessellation, which is the dual for the model of the standard double octahedral group 2 O h , generated by adding the new period 2 operation . However, the group O h can also be doubled with a different new period 2 operation to give a different double group 2 W h with periods of 2, 4, and 6 rather than the periods 2, 3, and 8 in 2 Oh .