Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
Study examines how changing regions affects planar graphs.
problem Effect of region crossing change on planar trivalent graphs.
method Investigation of region crossing changes on planar trivalent graphs.
result Effect of region crossing change on planar trivalent graphs.
In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For n≥1, we show that a (2n+1)-dimensional contact manifold M suppor…
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
Study of higher-dimensional contact manifolds and their properties.
problem Understanding contact manifolds in higher dimensions.
method Topological and symplectic methods, including open books.
result Many contact manifolds can be realized as iterated planar contact manifolds.
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…
We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Paper introduces a new invariant for planar knotoids.
problem Defining an invariant for planar knotoids.
method Using Gauss diagrams and transcendental functions.
result The invariant is a Vassiliev invariant of order one.
A graph is apex if it can be made planar by deleting a vertex, that is, ∃v such that G−v is planar. We define the related notions of edge apex, ∃e such that G−e is planar, and contraction apex, ∃e such that G/e is planar, as well as the analogues with a universal quantifier: ∀v…
Entire area-minimizing surfaces of density 2 are planar or quadratic
problem Classifying entire area-minimizing surfaces
method Using density and algebraic properties
result All such surfaces are planar or algebraic
The complement of a non-separating planar graph contains a K_n minor.
problem Characterizing the structure of complements of planar graphs.
method Analyzing the structure of complements of non-separating planar graphs and using examples to illustrate hypotheses.
result The order 2n-3 is the lowest possible for a non-separating planar graph whose complement contains a K_n minor.
We define a certain abstract planar algebra by generators and relations, study various aspects of its structure, and then identify it with Jones' spin planar algebra.
Sharp bounds for spanning tree entropy in planar lattices.
problem Estimating spanning tree entropy in planar lattice graphs.
method Using hyperbolic geometry and polyhedra volumes.
result Proved bounds are easy to compute and provide excellent estimates.
Study on planar graph braid groups' second homology.
problem Characterize the second homology of planar graph braid groups.
method Analyzing configuration spaces of planar graphs under specific operations.
result The second homology is generated by three specific graphs.
Paper defines a jellyfish algorithm for a specific subfactor planar algebra.
problem Diagrammatic presentation of generators and relations for E7 subfactor planar algebra. method Diagrammatic presentation and proof of well-definedness of jellyfish algorithm.
result Jellyfish algorithm is a well-defined surjection onto C for E7 subfactor planar algebra. Compact Special Weingarten surfaces with planar convex boundaries are disks.
problem Characterizing Special Weingarten surfaces with specific boundary conditions.
method Proved a Ros-Rosenberg theorem in the context of Special Weingarten surfaces.
result Compact Special Weingarten surfaces with planar convex boundaries are topological disks.
In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski 3− space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…
Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late 19th century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to re…
Determine lens spaces as closures of homology cobordisms over planar surfaces.
problem Identify conditions for lens spaces to be closures of homology cobordisms over planar surfaces.
method Use Chebotarev density theorem in the proof.
result Every lens space is represented as a closure of homology cobordism over a planar surface with three boundary components.
We present definitions and properties of conformal Killing, Killing and planarity forms on a Riemannian manifold and determine Tachibana, Killing and planarity numbers as an analog of the well known Betti numbers. We state some set of conditions to characterize these numbers. Moreover, we formulate the main results on …
Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.
problem Classify Stein fillings of planar contact 3-manifolds under constraints on their relative trisections.
method Partial classification of diffeomorphism types of fillings with relative trisections of genus at most 2.
result Partially classify the diffeomorphism types of Stein fillings with relative trisections of genus at most 2.
Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
problem Understanding the collapsibility of 2-complexes with specific curvature properties.
method Analyzing the fundamental groups and sectional curvatures of 2-complexes.
result Finite simply connected 2-complexes with nonpositive planar curvature are collapsible.
New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
In this paper we give two examples of sequences of embedded minimal planar domains in R3 which converge to singular laminations of R3. In contrast with the situation for embedded minimal disks, these examples do not arise from complete embedded minimal planar domains and highlight some of the su…
Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of …
This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.
problem Classifying planar-Rips complexes and their unit disk graphs.
method Simplicial classification, homotopy equivalence, and hereditary properties.
result Classification of planar-Rips complexes and unit disk graphs up to homotopy.
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…
We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic f…
Proves existence of planar curves with specific curvature.
problem Existence of planar closed curves with prescribed curvature.
method Variational methods, adding a parameter, monotonicity trick.
result Existence of planar closed curves with prescribed curvature for some curvature functions.
In his 1930 paper, Kuratowksi categorized planar graphs, proving that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K5, the complete graph on 5 vertices, or K3,3, the complete bipartite graph on six vertices. In their 2001 paper, Davis and Okun point out that…
Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of …
The paper proves nonexistence and existence results for minimal surfaces in R^4.
problem Proving nonexistence and existence of minimal surfaces in R^4.
method Analyzing complete minimal immersions with finite total curvature and embedded planar ends.
result Existence of embedded minimal spheres in R^4 with 3 embedded planar ends.
Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.
problem Understanding the relationship between unknotting numbers and crossing numbers of spatial embeddings of planar graphs.
method Analyzing specific examples of planar graphs and their spatial embeddings to find counterexamples.
result There exist planar graphs and their spatial embeddings where the unknotting number is greater than half the crossing number.
String graphs are closely related to planar graphs in terms of distances.
problem Understanding the relationship between string graphs and planar graphs.
method Proved quasi-isometric relationship between string graphs and planar graphs.
result String graphs are quasi-isometric to planar graphs.
Spatial graphs are decomposed into planar forests and braids.
problem Understanding the structure of spatial graphs in 3-space.
method Decomposition of spatial graphs into planar forests and braids.
result Every finite spatial graph is a connected sum of a planar graph and a braid.
The study extends Tutte's conflict graph concept to nonplanar graphs.
problem Understanding the structure of nonplanar graphs through conflict graphs.
method Defining a signed conflict graph for maximally planar subgraphs and analyzing their balance.
result For graphs with a flat embedding, every maximal planar subgraph has unbalanced conflict graphs if and only if the graph is intrinsically linked.
This paper defines ribbons and ribbon complexes in CW spaces and analyzes their topological properties.
problem Characterizing and analyzing topological structures in CW spaces.
method Introducing planar ribbons, ribbon complexes, and ribbon nerves in Alexandroff-Hopf-Whitehead CW spaces, and studying their topological properties.
result Characterization of ribbons and ribbon nerves by Betti numbers and homotopy types.
New quantum invariants for planar knotoids improve knot classification.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
We investigate half-lightlike submanifolds with planar normal sections of four dimensional pseudo Euclidean space. We obtain necessary and sufficient conditions for a half-lightlike submanifold of R24 such that it has degenerate or non-degenerate planar normal sections.
The paper explores knotoid invariants on the plane using quandles and cocycles.
problem Classifying planar knotoids due to their complexity and lack of invariants.
method Investigates equivalence of planar knotoids using quandle colorings and cocycle invariants.
result Introduces a new invariant called the triangular quandle cocycle invariant.
We define a computable topological invariant μ(γ) for generic closed planar regular curves γ, which gives an effective lower bound for the number of inflection points on a given generic closed planar curve. Using it, we classify the topological types of locally convex curves (i.e. closed planar regular curves witho…
New method for sensing non-planar surfaces using ERT.
problem Limited computational techniques for planar surfaces in ERT-based sensing skins.
method Generalized ERT to non-planar surfaces using Riemannian geometry.
result Feasibility and applicability of ERT-based sensing skins for non-planar geometries demonstrated.
Minimal moves for surfaces in 4D identified.
problem Classifying surfaces embedded in 4D space.
method Derived minimal generating set of planar moves.
result Identified minimal moves for surfaces in 4D.