Extends knot polynomial to knotted 4-valent graphs.
problem Constructing an invariant for knotted 4-valent graphs.
method Graphical calculus and Reidemeister moves for 4-valent graphs.
result Extension of polynomial to knotted 4-valent graphs.
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48 results for “4-valent graphs”
New polynomial for checkerboard-colorable 4-valent virtual graphs.
problem No specific problem stated; focuses on a new polynomial.
method Euler circuit expansion to assign polynomial to graphs.
result New combinatorial formulation of Kauffman-Jones polynomial.
The Kauffman-Vogel polynomials are three variable polynomial invariants of -valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented -valent rigid vertex graphs was given by using the Kauffman bracket and the Jones-Wenzl idempotent colored with . Bataineh, Elha…
Paper shows any link can be diagrammed with only triangles and quadrilaterals.
problem Representing links with only triangles and quadrilaterals.
method Examined link diagrams as 4-valent graphs on a sphere.
result Any link can be represented with only triangles and quadrilaterals.
We use 4-valent planar graphs and singular cobordisms (called foams) to construct an integral doubly-graded cohomology for tangles, and in particular for links, whose graded Euler characteristic yields the sl(n) link polynomial (for n > 3).
It is well known that surface-links in 4-space can be presented by diagrams on the plane of 4-valent spatial graphs with makers on the vertices, called marked graph diagrams. In this paper we extend the method of presenting surface-links by marked graph diagrams to presenting immersed surface-links. We also give some m…
The notion of a pseudoknot is defined as an equivalence class of knot diagrams that may be missing some crossing information. We provide here a topological invariant schema for pseudoknots and their relatives, 4-valent rigid vertex spatial graphs and singular knots, that is obtained by replacing unknown crossings or ve…
Motivated by his studies in knot theory V. Vassiliev introduced -graphs as regular 4-valent graph with a structure of pairs of opposite edges at each vertex. He conjectured the conditions under which -graph can be embedded into a plane respecting the the -structure at every vertex. The conjecture was proved by…
Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus with punctures for any and .
We generalize the colored Jones polynomial to -valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.
Call {\em i-hedrite} any 4-valent n-vertex plane graph, whose faces are 2-, 3- and 4-gons only and . The edges of an i-hedrite, as of any Eulerian plane graph, are partitioned by its {\em central circuits}, i.e. those, which are obtained by starting with an edge and continuing at each vertex by the edge oppo…
A marked graph diagram is a link diagram possibly with marked -valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa gave local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two…
The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal P^2-irreducible triangulation. In addition we present constraints upon the co…
By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken …
New knot invariant from 3-braids and 6-valent graphs.
problem Classical knot invariant construction.
method Using group and plat closure of braids, define a map to framed 6-valent graphs.
result Obtained a knot invariant valued in equivalence classes of graphs.
We construct potentially new manifolds homeomorphic but not diffeomorphic to and via rational blowdown surgery along certain -valent plumbing graphs. This way all the graph classes from \cite{weighted} have a represen…
In [3] we constructed the parity-biquandle bracket valued in {\em pictures} (linear combinations of -valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial. In the present paper we slightly change the notation of the parity-biquandle bracket and give exam…
Origamis' orbits are non-planar except for a few specific cases.
problem Determining the planarity of origamis' orbits under SL(2,Z) action.
method Analyzing 4-valent graphs from SL(2,Z) action on origamis in H(2).
result Most origamis' orbits are non-planar, with specific exceptions.
For a signed cyclic graph G, we can construct a unique virtual link L by taking the medial construction and convert 4-valent vertices of the medial graph to crossings according to the signs. If a virtual link can occur in this way then we say that the virtual link is graphical. In the article we shall prove that a virt…
We employ a solution of the Yang-Baxter equation to construct invariants for knot-like objects. Specifically, we consider a Yang-Baxter state model for the sl(n) polynomial of classical links and extend it to oriented singular links and balanced oriented 4-valent knotted graphs with rigid vertices. We also define a rep…
Improved linear upper bound for ribbonlength of knots.
problem Estimating the ribbonlength of knots and links.
method Using four-page open book decompositions and spanning trees of checkerboard graphs, constructing a four-page presentation with at most 2c(K) arcs.
result Proved that ribbonlength is bounded above by the four-page index, leading to the linear bound Rib(K) ≤ 2c(K).
Classifies doodles into prime and super prime types, describing them with doodle codes.
problem Classifying doodles into prime and super prime types.
method Using doodle codes to describe complementary regions and enumerate doodle diagrams.
result Super prime doodles have a Hamiltonian circuit.
This paper contains some more results on the topology of a nondegenerate action of on a compact connected -manifold when the action is totally hyperbolic (i.e. its toric degree is zero). We study the -action generated by a fixed vector of , that provides some results on t…
Aicardi's invariant is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends to colored singular links, showing it's stronger than HOMFLY-PT polynomial.
We describe a model of random links based on random 4-valent maps, which can be sampled due to the work of Schaeffer. We will look at the relationship between the combinatorial information in the diagram and the hyperbolic volume. Specifically, we show that for random alternating diagrams, the expected hyperbolic volum…
Characterizes arithmetic and commensurable links in curved surfaces.
problem Classifying arithmetic and commensurable links in curved surfaces.
method Combines symmetry arguments, combinatorial geometry, and number-theoretic data.
result Characterizes arithmetic and commensurable right-angled tiling links.
A plane curve is a knot diagram in which each crossing is replaced by a 4-valent vertex, and so are dual to a subset of planar quadrangulations. The aim of this paper is to introduce a new tool for sampling diagrams via sampling of plane curves. At present the most efficient method for sampling diagrams is rejection sa…
Quantum theory of curved tetrahedrons yields quantum group intertwiners.
problem Quantum geometry of curved tetrahedrons and their intertwiners.
method Combinatorial quantization of tetrahedron phase space, relating to SU(2) flat connections.
result Physical Hilbert space coincides with Uq(su(2)) intertwiners, consistent with LQG area spectrum.
In the present paper, we define an invariant of free links valued in a free product of some copies of . In \cite{Ma2} the second named author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not…
We consider the problem of counting and of listing topologically inequivalent "planar" {4-valent} maps with a single component and a given number n of vertices. This enables us to count and to tabulate immersions of a circle in a sphere (spherical curves), extending results by Arnold and followers. Different options wh…
Line graph transformation aids graph isomorphism tests by excluding challenging graph properties.
problem Limited theoretical understanding of line graph transformation's impact on GNN models.
method Examined CFI and strongly regular graphs, showing line graph transformation helps WL tests distinguish these graphs.
result Line graph transformation aids WL tests in distinguishing challenging graph properties.
Proposes MGMN for end-to-end graph similarity learning.
problem Lack of cross-level interactions in graph similarity learning.
method Multi-level graph matching network (MGMN) combining node-graph matching and siamese graph neural networks.
result MGMN outperforms state-of-the-art models on graph-graph classification and regression tasks.
Graphons of Line Graphsstat.ML
The paper explores graphons of line graphs from sparse finite graphs.
problem Estimating graph limits from sparse finite graphs.
method Mapping finite graphs to their line graphs and analyzing graphs with the square-degree property.
result Graphons of line graphs can distinguish between sparse graphs like star graphs and superlinear preferential attachment graphs.
MxPool learns graph features from diverse graphs using a hierarchical structure.
problem Learning graph features from diverse graphs with varying properties and sizes.
method MxPool uses a multiplex structure with multiple graph convolution/pooling networks in a hierarchical learning structure.
result MxPool outperforms state-of-the-art methods on graph classification benchmarks.
Study the geometry of graph product extension graphs.
problem Properties of graph products.
method Introduce and study the extension graph of graph products of groups.
result Extension graph is isomorphic to crossing graph of a quasi-median graph and exhibits asymptotic dimension similar to quasi-trees.
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.
Customized-GNN generates model-specific for each graph.
problem Graphs in the same dataset have distinct structures.
method Proposes Customized-GNN framework to generate model-specific for each graph.
result Demonstrates effectiveness on various graph classification benchmarks.
Graph embedding leaks sensitive graph properties and subgraphs.
problem Privacy risks in graph embedding sharing.
method Three inference attacks and a defense mechanism.
result High accuracy in inferring graph properties and subgraphs.
Characterizes graphs with leveled embeddings and introduces new graph invariants.
problem Understanding the properties of leveled embeddings in spatial graphs.
method Characterization of graphs with leveled embeddings, introduction of new invariants.
result Characterization of graphs with low level number and determination of specific invariants for complete graphs and complete bipartite graphs.
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
Two new methods improve graph embedding without needing a complete graph structure.
problem Graph autoencoders' performance depends on the adjacency matrix quality.
method BAGE and VBAGE: unsupervised graph embedding via adaptive graph learning.
result The methods expand GAEs' applicability to datasets without graph structure.
We define a pseudo-inverse for line graphs using linear integer programming.
problem Not all graphs have a corresponding root graph, making the line graph operation non-invertible.
method Propose a linear integer program to edit the smallest number of edges in the line graph to recover a root graph.
result The pseudo-inverse operation is well-behaved and works in practice as shown by empirical experiments.
Develops method to create non-Abelian Ricci-flat graphs via bundles.
problem Creating non-Abelian Ricci-flat graphs.
method Develops systematic way via graph bundles with constraints.
result Non-trivial graph bundles are not isomorphic to product of base and fiber.
New method uses graph generative models for graph classification.
problem Graph classification for non-relational i.i.d. data.
method Derive classification formulas from GGM, train generative graph auto-encoder model.
result New conditional ELBO for training graph auto-encoder model.
MathNet uses wavelets for graph representation and learning.
problem Graph Neural Networks (GNNs) for graph classification and regression.
method Multiresolution Haar-like wavelets, graph convolution, and pooling.
result MathNet achieves notable accuracy gains on graph classification and regression tasks.
Unified framework for graph coarsening using node features and graph matrices.
problem Dimensionality reduction of large graphs while preserving node features.
method Optimization-based framework that unifies graph learning and dimensionality reduction.
result The learned coarsened graph is ε-similar to the original graph, where ε is a small positive number.
A fast graph embedding method for large graphs.
problem Efficiently embedding large graphs for various applications.
method One-hot graph encoder embedding with linear complexity.
result Graph encoder embedding is approximately normally distributed and converges to its mean.
Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.