Surfaces Of Arbitrary Genus Research Articles

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Compatible intrinsic triangulations

Finding distortion-minimizing homeomorphisms between surfaces of arbitrary genus is a fundamental task in computer graphics and geometry processing. We propose a simple method utilizing intrinsic triangulations, operating directly on the original surfaces without going through any intermediate domains such as a plane or a sphere. Given two models A and B as triangle meshes, our algorithm constructs a Compatible Intrinsic Triangulation (CIT), a pair of intrinsic triangulations over A and B with full correspondences in their vertices, edges and faces. Such a tessellation allows us to establish consistent images of edges and faces of A's input mesh over B (and vice versa) by tracing piecewise-geodesic paths over A and B. Our algorithm for constructing CITs, primarily consisting of carefully designed edge flipping schemes, is empirical in nature without any guarantee of success, but turns out to be robust enough to be used within a similar second-order optimization framework as was used previously in the literature. The utility of our method is demonstrated through comparisons and evaluation on a standard benchmark dataset.

Classification of exceptional nodal topologies protected by PT symmetry

Exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, and parity-time ($\mathcal{PT}$) symmetry, reflecting balanced gain and loss in photonic systems, are paramount concepts in non-Hermitian systems. We here complete the topological classification of exceptional nodal degeneracies protected by $\mathcal{PT}$ symmetry in up to three dimensions and provide simple example models whose exceptional nodal topologies include previously overlooked possibilities such as second-order knotted surfaces of arbitrary genus, third-order knots and fourth-order points.

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.

Combinatorial Construction of Seamless Parameter Domains

AbstractThe problem of seamless parametrization of surfaces is of interest in the context of structured quadrilateral mesh generation and spline‐based surface approximation. It has been tackled by a variety of approaches, commonly relying on continuous numerical optimization to ultimately obtain suitable parameter domains. We present a general combinatorial seamless parameter domain construction, free from the potential numerical issues inherent to continuous optimization techniques in practice. The domains are constructed as abstract polygonal complexes which can be embedded in a discrete planar grid space, as unions of unit squares. We ensure that the domain structure matches any prescribed parametrization singularities (cones) and satisfies seamlessness conditions. Surfaces of arbitrary genus are supported. Once a domain suitable for a given surface is constructed, a seamless and locally injective parametrization over this domain can be obtained using existing planar disk mapping techniques, making recourse to Tutte's classical embedding theorem.

Low Dimensional Matrix Representations for Noncommutative Surfaces of Arbitrary Genus

In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated in detail and graph representations are used in order to understand the structure of non-zero matrix elements. In particular, for arbitrary genus greater than one, we explicitly construct classes of irreducible two and three dimensional representations. The existence of representations crucially depends on the analytic structure of the polynomial defining the surface as a level set in mathbb {R}^{3}.

Modular orbits at higher genus

We extend the modular orbits method of constructing a two-dimensional orbifold conformal field theory to higher genus Riemann surfaces. We find that partition functions on surfaces of arbitrary genus can be constructed by a straightforward generalization of the rules that one would apply to the torus. We demonstrate how one can use these higher genus objects to compute correlation functions and OPE coefficients in the underlying theory. In the case of orbifolds of free bosonic theories by subgroups of continuous symmetries, we can give the explicit results of our procedure for symmetric and asymmetric orbifolds by cyclic groups.

A duality in two-dimensional gravity

We demonstrate an equivalence between two integrable flows defined in a polynomial ring quotiented by an ideal generated by a polynomial. This duality of integrable systems allows us to systematically exploit the Korteweg-de Vries hierarchy and its tau-function to propose amplitudes for non-compact topological gravity on Riemann surfaces of arbitrary genus. We thus quantize topological gravity coupled to non-compact topologica matter and demonstrate that this phase of topological gravity at N = 2 matter central charge larger than three is equivalent to the phase with matter of central charge smaller than three.

Knot invariants from Virasoro related representation and pretzel knots

We remind the method to calculate colored Jones polynomials for the plat representations of knot diagrams from the knowledge of modular transformation (monodromies) of Virasoro conformal blocks with insertions of degenerate fields. As an illustration we use a rich family of pretzel knots, lying on a surface of arbitrary genus g, which was recently analyzed by the evolution method. Further generalizations can be to generic Virasoro modular transformations, provided by integral kernels, which can lead to the Hikami invariants.

The Las Vergnas polynomial for embedded graphs

The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion–contraction relations, its specialisation to cellularly embedded graphs does not. Here we extend the Las Vergnas polynomial to graphs in pseudo-surfaces. We show that in this setting we can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobás–Riordan polynomial, and the Krushkal polynomial. We also take this opportunity to extend some of Las Vergnas’ results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus.

On flexible polyhedral surfaces

We construct a closed orientable polyhedral surface of arbitrary genus that is embedded in three-dimensional Euclidean space and admits a one-parameter bending under which all its handles bend. This surface admits no other bendings. We also construct a flexible closed nonorientable polyhedral surface of arbitrary genus such that all its handles and Mobius strips bend during its bending.

Classification of topological symmetry sectors on anyon rings

The golden chain with antiferromagnetic interaction is an anyonic system of particular interest as when all anyons are confined to the chain, it is readily stabilized against fluctuations away from criticality. However, additional local scaling operators have recently been identified on the disk which may give rise to relevant fluctuations in the presence of free charges. Motivated by these results for Fibonacci anyons, this paper presents a systematic method of identifying all topological sectors of local scaling operators for critical anyon rings of arbitrary winding number on surfaces of arbitrary genus, extending the original classification scheme proposed by Feiguin et al. Using the new scheme, it is then shown that for the golden chain, additional relevant scaling operators exist on the torus which are equivalent to those detected on the disk and which may disrupt the stability of the critical system. Protection of criticality against perturbations generated by these additional scaling operators can be achieved by suppressing the exchange of charge between the anyon ring and the rest of the manifold.

Connecting face hitting sets in planar graphs

We show that any face hitting set of size n of a connected planar graph with a minimum degree of at least 3 is contained in a connected subgraph of size 5 n − 6 . Furthermore we show that this bound is tight by providing a lower bound in the form of a family of graphs. This improves the previously known upper and lower bound of 11 n − 18 and 3 n respectively by Grigoriev and Sitters. Our proof is valid for simple graphs with loops and generalizes to graphs embedded in surfaces of arbitrary genus.

New Maximal Surfaces in Minkowski 3-Space with Arbitrary Genus and Their Cousins in de Sitter 3-Space

Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean 3-space given by the third and fourth authors in a previous paper. In this paper, we discuss singularities and several global properties of maximal surfaces, and give explicit examples of such surfaces of arbitrary genus. When the genus is one, our examples are embedded outside a compact set. Moreover, we deform such examples to CMC-1 faces (mean curvature one surfaces with admissible singularities in de Sitter 3-space) and obtain “cousins of those maximal surfaces.

Geometry Images of Arbitrary Genus in the Spherical Domain

Abstract While existing spherical parameterization algorithms are limited to genus‐0 geometrical models, we believe a wide class of models of arbitrary genus can also benefit from the spherical domain. We present a complete and robust pipeline that can generate spherical geometry images from arbitrary genus surfaces where the holes are explicitly represented. The geometrical model, represented as a triangle mesh, is first made topologically equivalent to a sphere by cutting each hole along its generators, thus performing genus reduction. The resulting genus‐0 model is then parameterized on the sphere, where it is resampled in a way to preserve connectivity between holes and to reduce the visual impact of seams due to these holes. Knowing the location of each pair of boundary components in parametric space, our novel sampling scheme can automatically choose to scale down or completely eliminate the associated hole, depending on geometry image resolution, thus lowering the genus of the reconstructed model. We found our method to scale better than other geometry image algorithms for higher genus models. We illustrate our approach on remeshing, level‐of‐detail rendering, normal mapping and topology editing.

Stability of rank 2 vector bundles along isomonodromic deformations

We are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact Riemann surfaces, which are provided with irreducible meromophic tracefree connections. In the case of a logarithmic connection on the Riemann sphere, such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles, according to a result of A. Bolibruch. In the general case of meromorphic connections over Riemann surfaces of arbitrary genus, we prove that the vector bundle will be semi-stable, up to a small isomonodromic deformation. More precisely, the vector bundle underlying the universal isomonodromic deformation is generically semi-stable along the deformation, and even maximally stable. For curves of genus g ≥ 2, this result is non-trivial even in the case of non-singular connections.

Schnyder woods for higher genus triangulated surfaces (abstract)

Abstract We study a well known characterization of planar graphs, also called Schnyder wood or Schnyder labelling, which yields a decomposition into vertex spanning trees. The goal is to extend previous algorithms and characterizations designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. We define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and colouration of the edges that extends the one proposed by Schnyder for the planar case. As a by-product we show how to characterize our edge coloration in terms of genus g maps.

Approximate convex decomposition of polyhedra and its applications

Decomposition is a technique commonly used to partition complex models into simpler components. While decomposition into convex components results in pieces that are easy to process, such decompositions can be costly to construct and can result in representations with an unmanageable number of components. In this paper we explore an alternative partitioning strategy that decomposes a given model into “approximately convex” pieces that may provide similar benefits as convex components, while the resulting decomposition is both significantly smaller (typically by orders of magnitude) and can be computed more efficiently. Indeed, for many applications, an approximate convex decomposition ( acd) can more accurately represent the important structural features of the model by providing a mechanism for ignoring less significant features, such as surface texture. We describe a technique for computing acds of three-dimensional polyhedral solids and surfaces of arbitrary genus. We provide results illustrating that our approach results in high quality decompositions with very few components and applications showing that comparable or better results can be obtained using acd decompositions in place of exact convex decompositions ( ecd) that are several orders of magnitude larger.

Homological error correction: Classical and quantum codes

We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly. Not every classical code is homological, but we find a family of classical homological codes saturating the Hamming bound. In the quantum case, we show that for nonorientable surfaces it is impossible to construct homological codes based on qudits of dimension D>2, while for orientable surfaces with boundaries it is possible to construct them for arbitrary dimension D. We give a method to obtain planar homological codes based on the construction of quantum codes on compact surfaces without boundaries. We show how the original Shor’s 9-qubit code can be visualized as a homological quantum code. We study the problem of constructing quantum codes with optimal encoding rate. In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality. For homological quantum codes on surfaces of arbitrary genus we also construct a family of codes asymptotically attaining the maximum possible encoding rate. We provide the tools of homology group theory for graphs embedded on surfaces in a self-contained manner.

Topological quantum error correction with optimal encoding rate

We prove the existence of topological quantum error correcting codes with encoding rates $k/n$ asymptotically approaching the maximum possible value. Explicit constructions of these topological codes are presented using surfaces of arbitrary genus. We find a class of regular toric codes that are optimal. For physical implementations, we present planar topological codes.