We study combinatorial modulus on boundaries of hyperbolic Coxeter groups. We give new examples of hyperbolic groups whose boundary satisfies a combinatorial version of the Loewner property, and prove Cannon's conjecture for Coxeter groups. We also establish some connections with l^p cohomology.
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48 results for “combinatorial properties”
Combinatorial proof of grid homology properties.
problem Properties of double-point enhanced grid homology.
method Purely combinatorial proof, extended to coefficients.
result Skein exact sequence obeyed by grid homology.
Sarkar and Wang have given a combinatorial algorithm for computing Heegaard Floer homology and Plamenevskaya has improved their method to compute Ozsvath-Szabo invariant. In this paper, applying the combinatorial method to stabilizations of an open book, we prove basic properties of Ozsvath-Szabo invariant.
Study explores properties of bipartite knots.
problem None explicitly stated; focuses on properties of bipartite knots.
method Exploration of combinatorial structure.
result Rich combinatorial structure of bipartite knots.
New fractal spaces not quasisymmetric to Loewner spaces discovered.
problem Finding new fractal spaces not quasisymmetric to Loewner spaces.
method Introduced iterated graph systems (IGS) to create new fractal spaces.
result Disproved Kleiner's conjecture about self-similar fractals.
Detecting topological properties of Markov compacta with combinatorial properties of their diagramsmath.GT
We develop a formalism that allows us to describe Markov compacta with finite sets of diagrams that are building blocks of the entire sequence. This encodes complex, continuous spaces with discrete collections of combinatorial objects. We show that topological properties of the limit (such as -connectedness, local $…
In this survey on combinatorial properties of triangulated manifolds we discuss various lower bounds on the number of vertices of simplicial and combinatorial manifolds. Moreover, we give a list of all known examples of vertex-minimal triangulations.
In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite fami…
New property helps show many knot fillings are not left-orderable.
problem Non-left-orderability of knot groups after Dehn filling.
method Introducing conjugate-slope property to study knot groups.
result The conjugate-slope property implies non-left-orderability of many knot fillings.
In this article, we discuss the quasiconformal structure of boundaries of right-angled hyperbolic buildings using combinatorial tools. In particular we exhibit some examples of buildings of dimension 3 and 4 whose boundaries satisfy the combinatorial Loewner property. This property is a weak version of the Loewner prop…
In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove …
We consider a stabilized version of hat Heegaard Floer homology of a 3-manifold Y (i.e. the U=0 variant of Heegaard Floer homology for closed 3-manifolds). We give a combinatorial algorithm for constructing this invariant, starting from a Heegaard decomposition for Y, and give a combinatorial proof of its invariance pr…
We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for the combinatorial one and that, moreover, the same results hold for a more gene…
Formula derived for a magnetic line invariant.
problem Solving MHD problems with two-scale mean fields.
method Combinatorial definition of invariant for three-component links.
result Proven formula for invariant verified for simple cases.
We study the fundamental properties of curvature in groupoids within the framework of synthetic differential geometry. As is usual in synthetic differential geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
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 t…
Detects right-veering properties in open books using combinatorial methods.
problem Determining the right-veering property of contact structures.
method Combinatorial approach to detect left-veering arcs in open books.
result Existence of an algorithm to detect right-veering for compact surfaces.
The paper classifies compact hyperbolic Coxeter polytopes and improves upper bounds.
problem Classifying compact hyperbolic Coxeter polytopes and understanding their combinatorial properties.
method Study of -products of Lannér diagrams, proving superhyperbolic properties, and analyzing Lannér subdiagrams.
result Improved upper bounds on the dimension of compact hyperbolic Coxeter polytopes.
The study generalizes origamis to flat surfaces, exploring their combinatorial and geometric properties.
problem Understanding the geometric and combinatorial properties of flat surfaces.
method Developing a system of linear equations to represent flat surfaces and studying their Veech groups.
result Veech groups of certain flat surfaces are included under a specific covering relation.
We study quasi-isometry invariants of Gromov hyperbolic spaces, focussing on the l_p-cohomology and closely related invariants such as the conformal dimension, combinatorial modulus, and the Combinatorial Loewner Property. We give new constructions of continuous l_p-cohomology, thereby obtaining information about the l…
We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from work of Johnson and Thompson, called thin position. Thin position is defined usin…
In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a compl…
In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.
We introduce a combinatorial curvature flow for PL metrics on compact triangulated 3-manifolds with boundary consisting of surfaces of negative Euler characteristic. The flow tends to find the complete hyperbolic metric with totally geodesic boundary on a manifold. Some of the basic properties of the combinatorial flow…
In this paper we present the Ricci curvature on cell-complexes and show the Gauss-Bonnnet type theorem on graphs and 2-complex that decomposes closed surface. The defferential forms on a cell complex is defined as linear maps on chain complex, and Laplacian operates this defferential forms. Then we construct the Bochne…
The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.
problem Finding piecewise constant curvature metrics on surfaces with prescribed combinatorial curvatures.
method Combinatorial curvature flows, including Ricci flow and Calabi flow, are applied to deform Glickenstein's discrete conformal structures.
result The solution of the combinatorial Ricci flow can be uniquely extended and converges exponentially fast for any initial value under certain conditions.
Introduces a new manifold from a graph subgraph.
problem None explicitly stated in the abstract.
method Defined geometrically from a combinatorial subgraph of the Hoffman-Singleton graph.
result Geometric properties of the new manifold presented.
The paper computes special values of combinatorial zeta functions to reveal topological properties of manifolds.
problem Counting geodesic paths in triangulations to infer topological invariants.
method Random walks on higher-dimensional skeletons of triangulations.
result Recovery of Betti numbers and linking numbers of manifolds.
A property that characterizes Euler characteristic among invariants of combinatorial manifoldsmath.GT
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique inv…
New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.
problem Creating new discrete conformal structures on surfaces with boundary.
method Introducing new discrete conformal structures, proving global rigidity using variational principles, and introducing combinatorial curvature flows.
result Global rigidity of new discrete conformal structures and effective algorithms for constructing hyperbolic metrics.
Developed a real sutured Heegaard Floer theory.
problem Real sutured manifolds and their Floer homology.
method Real nice diagrams, combinatorial computation.
result New structural properties distinguishing from sutured Floer homology.
The present paper produces examples of Gauss diagram formulae for virtual knot invariants which have no analogue in the classical knot case. These combinatorial formulae contain additional information about how a subdiagram is embedded in a virtual knot diagram. The additional information comes from the second author's…
Paper resolves spherical curvature flow problem.
problem Existence of ideal circle patterns in spherical background geometry.
method Introduces a combinatorial geodesic curvature flow in spherical background geometry.
result Characterizes sufficient and necessary conditions for flow convergence.
This paper applies combinatorial testing to machine learning for robust model performance.
problem Identifying robust machine learning models using test and training sets.
method Adapting combinatorial interaction testing for machine learning, focusing on simple features.
result Combinatorial coverage can enhance model performance and robustness.
A framework for generating 3D shapes by sequentially assembling primitives.
problem Combinatorial complexity in generating 3D shapes.
method Bayesian optimization for efficient exploration and exploitation of feasible combinations.
result Successfully generates realistic combinatorial 3D shapes.
The paper reveals a property of chromatic homology for complete graphs.
problem Understanding the chromatic homology of complete graphs.
method Introduced a combinatorial description of enhanced states and used it to analyze the homology.
result Showed a splitting property of the chromatic homology for certain graphs.
Link Floer homology is an invariant for links defined using a suitable version of Lagrangian Floer homology. In an earlier paper, this invariant was given a combinatorial description with mod 2 coefficients. In the present paper, we give a self-contained presentation of the basic properties of link Floer homology, incl…
We extend the cobordism based categorification of the virtual Jones polynomial to virtual tangles. This extension is combinatorial and has semi-local properties. We use the semi-local property to prove an applications, i.e. we give a discussion of Lee's degeneration of virtual homology.
In this paper we extend the classical theory of combinatorial manifolds to the non-homogeneous setting. NH-manifolds are polyhedra which are locally like Euclidean spaces of varying dimensions. We show that many of the properties of classical manifolds remain valid in this wider context. NH-manifolds appear naturally w…
Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groupsmath.SG
A proof of non-existence of Lagrangian embeddings of the Klein bottle K in \CP^2 is given. We exploit the existence of a special embedding of K in a symplectic Lefschetz pencil on \CP^2 and study its monodromy. As the main technical tool, we develop the theory of mapping class groups, considered as quotients of special…
A new deep learning framework for topological data.
problem Developing models for data on complex topological domains.
method Introducing combinatorial complexes and developing attention-based CCNNs.
result CCNNs outperform existing models in tasks involving mesh shape analysis and graph learning.
Paper studies geometric and combinatorial properties of circular snakes.
problem Exploring geometric and combinatorial properties of circular snakes.
method Definition and investigation of outer Lipschitz geometry, decomposition of Valette link, construction of combinatorial objects, weakly outer Lipschitz classification.
result Existence of canonical decomposition and necessary/sufficient criteria for removing segments or Hölder triangles.
A new algorithm balances exploration and exploitation in online decision-making.
problem Balancing exploration and exploitation in online decision-making.
method Proposed -UCB algorithm incorporating conservative mechanism.
result Proved n-step upper regret bound for two situations.
We count meromorphic differentials with fixed residues and poles of fixed orders.
problem Counting meromorphic differentials with fixed residues and poles of fixed orders.
method Intersection theory on compactified moduli spaces of differentials.
result Complete solution to the problem with interesting combinatorial properties.
It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result to the abstract setting of an infinite EI category satisfying certain combinatorial conditions.
In this survey we discuss how geometric methods can be used to study topological properties of 3-manifolds such as their Heegaard genus or the rank of their fundamental group. On the other hand, we also discuss briefly some results relating combinatorial descriptions and geometric properties of hyperbolic 3-manifolds.
In this paper we study a new combinatorial invariant of simple polytopes, which comes from toric topology. With each simple n-polytope P with m facets we can associate a moment-angle complex Z_P with a canonical action of the torus T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on Z_P. The…
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.