arXiv research

Characterizes 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.

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 $n\geq 1$, we show that a $(2n+1)$-dimensional contact manifold $M$ suppor…

New inequalities for planar convex domains' Laplacian eigenvalues.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

Study of higher-dimensional contact manifolds and their properties.

Proves planar graphs' configuration spaces have highest topological complexity.

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.

Paper introduces a new invariant for planar knotoids.

A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that $G/e$ is planar, as well as the analogues with a universal quantifier: $\forall v$…

Entire area-minimizing surfaces of density 2 are planar or quadratic

The complement of a non-separating planar graph 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.

Study on planar graph braid groups' second homology.

Paper defines a jellyfish algorithm for a specific subfactor planar algebra.

Compact Special Weingarten surfaces with planar convex boundaries are 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…

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.

Study Stein fillings of planar contact 3-manifolds with relative trisection genus 2.

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.

In this paper we give two examples of sequences of embedded minimal planar domains in $\mathbb{R}^3$ which converge to singular laminations of $\mathbb{R}^3$. 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.

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.

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 $K_5$, the complete graph on 5 vertices, or $K_{3,3}$, 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.

Spatial embeddings of planar graphs can have higher unknotting numbers than crossing numbers.

String graphs are closely related to planar graphs in terms of distances.

Spatial graphs are decomposed into planar forests and braids.

The study extends Tutte's conflict graph concept to nonplanar graphs.

New quantum invariants for planar knotoids improve knot classification.

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 $R_{2}^{4}$ such that it has degenerate or non-degenerate planar normal sections.

The paper explores knotoid invariants on the plane using quandles and cocycles.

This article introduces planar ribbons, Vergili ribbon complexes and ribbon nerves in Alexandroff-Hopf-Whitehead CW (Closure finite Weak) topological spaces. A {\em planar ribbon} (briefly, {ribbon}) in a CW space is the closure of a pair of nesting, non-concentric filled cycles that includes the boundary but does not …

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.

We give a complete description of all order 1 invariants of planar curves.

Minimal moves for surfaces in 4D identified.