arXiv research

Sharp bounds for spanning tree entropy in planar lattices.

Alternating links bound rational homology balls if their chessboard lattice is cubiquitous.

A lamination of a graph embedded on a surface is a collection of pairwise disjoint non-contractible simple closed curves drawn on the graph. In the case when the surface is a sphere with three punctures (a.k.a. a pair of pants), we first identify the lamination space of a graph embedded on that surface as a lattice pol…

Characterizes minor-minimal separating projective planar graphs and their generalizations.

Maximizes mixing efficiency in surface braids.

Study examines how changing regions affects planar graphs.

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

Study on planar graph braid groups' second homology.

New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.

The complement of a non-separating planar graph contains a K_n minor.

Study on planar graphs in Poincare model of hyperbolic geometry.

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

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

Spatial graphs are decomposed into planar forests and braids.

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…

Proves planar graphs' configuration spaces have highest topological complexity.

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: $\forall v$…

Automorphisms of fine curve graphs match surface homeomorphisms for planar surfaces.

Study classifies Halin graphs with positive curvature.

The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.

We show that given a trivalent graph in $S^3$, either the graph complement contains an essential almost meridional planar surface or thin position for the graph is also bridge position. This can be viewed as an extension of a theorem of Thompson to graphs. It follows that any graph complement always contains a useful p…

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

Planar multilinks prove rational singularities in surface geometry.

Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.

We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $λ$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -- lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable ``ultra-local'' Poisson bracket algebra defined on di…

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize …

We show that all nontrivial embeddings of planar graphs on the torus contain a nontrivial knot or a nonsplit link. This is equivalent to showing that no minimally knotted planar spatial graphs on the torus exist that contain neither a nontrivial knot nor a nonsplit link all of whose components are unknots.

In \cite{4} Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph $G$ a polynomial, denoted $[G]$, in three variables, $A$, $B$ and $a$, satisfies the skein relation: $$ [\psdiag{2}{6}{overcross}]=…

Approximates cycles in planar and bounded-genus graphs.

This paper classifies planar-Rips complexes and their unit disk graphs up to homotopy.

We prove that the spectral gap of a finite planar graph $X$ is bounded by $λ_1(X)\le C(\frac{\log(\diam X)}{\diam X})^2$ where $C$ depends only on the degree of $X$. We then give a sequence of such graphs showing the the above estimate cannot be improved. This yields a negative answer to a question of Benjamini and Cur…

The study connects lattices, Garside structures, and weakly modular graphs.

New method realizes planar graphs as Reeb graphs of algebraic functions.

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…

New bounds on diameters and generators for specific lattices and graphs.

Generalizes Kauffman's clock theorem to surfaces.

The paper refines transformations of lattice diagrams and introduces dotted diagrams.

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is …

Based on \cite{DH94}, we introduce a bijective correspondence between first order differential calculi and the graph structure of the symmetric lattice that allows one to encode completely the interconnection structure of the graph in the exterior derivative. As a result, we obtain the Grassmannian character of the lat…

Discrete knot theory models use lattice-filtered graphs to detect merging knot components.

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tre…

Temperley-Lieb algebras have been generalized to sl(3) web spaces. Since a cubic bipartite planar graph with suitable directions on edges is a web, the quantum sl(3) invariants naturally extend to all cubic bipartite planar graphs. First we completely classify them as a connected sum of primes webs. We also provide a m…

Origamis' orbits are non-planar except for a few specific cases.

We study the atomic embeddability testing problem, which is a common generalization of clustered planarity (c-planarity, for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995…

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and o…

Moduli space linked to Tait colorings of planar graphs.

We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…