Paper generalizes Yamada polynomial to virtual spatial graphs.
problem Generalizing classical knot theory to virtual spatial graphs.
method Topological definition and combinatorial approach for virtual spatial graphs.
result Generalized Yamada polynomial defined and proven invariant.
Two natural generalizations of knot theory are the study of spatially embedded graphs, and Kauffman's theory of virtual knots. In this paper we combine these approaches to begin the study of virtual spatial graphs.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
Graphoids are topological invariants of virtual graph diagrams.
problem Understanding knotted graphs with open ends in proteins and simplifying virtual spatial graphs.
method Topological interpretations of graphoids using graph Reidemeister moves.
result Virtual graphoids are useful for studying knotted graphs and simplifying spatial graphs.
Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results. We state the definitions, provide some examples, and survey the known results. W…
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several ex…
In order to apply quantum topology methods to nonplanar graphs, we define a planar diagram category that describes the local topology of embeddings of graphs into surfaces. These \emph{virtual graphs} are a categorical interpretation of ribbon graphs. We describe an extension of the flow polynomial to virtual graphs, t…
We prove Alexander- and Markov-type theorems for virtual spatial trivalent graphs and virtual trivalent braids. We provide two versions for the Markov-type theorem: one uses an algebraic approach similar to the case of classical braids and the other one is based on L-moves.
IGNNK uses GNN for spatiotemporal kriging, improving scalability and transferability.
problem Efficiently recovering signals for unsampled locations in spatiotemporal data.
method Developed an Inductive Graph Neural Network Kriging (IGNNK) model to learn spatial message passing.
result IGNNK effectively learns spatial message passing and can be transferred to new graph structures.
New equivalence relation on ribbon graphs connects to virtual links.
problem Understanding virtual links through ribbon graphs.
method Introducing a new equivalence relation on ribbon graphs.
result Correspondence between virtual links and ribbon graphs.
The paper generalizes virtual knot theory using multiple types of virtual crossings.
problem Generalizing virtual knot theory to include multiple types of virtual crossings.
method Starting with graph theory, the paper reviews previous work and then constructs multi-virtual knots and links.
result The multiplicity of virtual crossings allows for a broader application of the Penrose evaluation to all trivalent 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.
Twisted graph diagrams are virtual graph diagrams with bars on edges. A bijection between abstract graph diagrams and twisted graph diagrams is constructed. Then a polynomial invariant of Yamada-type is developed which provides a lower bound for the virtual crossing number of virtual graph diagrams.
Virtual knots with same writhe polynomial have equivalent intersection graphs.
problem Equivalence of intersection graphs for virtual knots.
method Proved equivalence through writhe polynomial.
result Intersection graphs of virtual knots with the same writhe polynomial are equivalent.
Proves minimal crossing diagrams for specific spatial graphs.
problem Proving minimal crossing diagrams for spatial graphs.
method Analyzing adequate diagrams and replacing vertices and edges.
result All 1-vertex spatial graphs with adequate diagrams have minimal crossing number.
New invariants distinguish spatial graphs not previously possible.
problem Distinguishing spatial graphs using Dehn colorings.
method Developed vertex-weight invariants based on Dehn colorings.
result Found spatial graphs distinguishable by vertex-weight invariants.
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…
Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
problem Unknottability of spatial graphs by region crossing changes.
method Region crossing changes to switch over/under relations within regions of spatial graph diagrams.
result Spatial graphs of non-Eulerian or proper Eulerian planar graphs are unknottable by region crossing changes.
Learning representation for graph classification turns a variable-size graph into a fixed-size vector (or matrix). Such a representation works nicely with algebraic manipulations. Here we introduce a simple method to augment an attributed graph with a virtual node that is bidirectionally connected to all existing nodes…
Grid homology theory for spatial graphs extends skein sequence.
problem No specific problem stated; focuses on extending a sequence.
method Defined grid homology theory for spatial graphs and extended skein sequence.
result Skein exact sequence extended to grid homology for spatial graphs.
Review of invariants for spatial graphs.
problem No specific problem stated; review of existing invariants.
method Combinatorial and polynomial invariants of spatial graphs.
result Overview of Alexander polynomial, fundamental quandle, and Yamada polynomial.
Defines concordance for spatial graphs and proves sliceness equivalence.
problem Understanding concordance and sliceness for spatial graphs.
method Smooth definitions and linking number conditions.
result Sliceness of a spatial graph is equivalent to a condition on linking numbers and a link.
Spatial graphs can be unknotted with region crossing changes.
problem Unknotted spatial graphs composed of theta-curves.
method Region crossing changes on regions of theta-curves.
result Spatial graphs of theta-curves can be unknotted.
Survey on spatial graphs with few vertices and edges.
problem Topology of spatial graphs with limited vertices and edges.
method Survey and focus on Brunnian θ-graphs.
result Survey reveals insights into spatial graphs.
New method reduces spatial graphs while preserving their topological features.
problem Finding a smaller spatial graph with the same structure.
method Topological spatial graph coarsening approach based on triangle-aware graph filtration.
result Significant reduction in graph size while preserving topological information.
Algorithm determines spatial graph isomorphism with vertex, edge colorings and orientations.
problem Algorithmic recognition of spatial graphs with various colorings and orientations.
method Proved existence of an algorithm for isomorphic spatial graphs, decomposed into canonical blocks, and applied Haken and Matveev's result.
result Algorithmic recognition of spatial graphs with colorings and orientations.
New results on splitting tangles and spatial graphs.
problem Issues with previous splitting results about tangles and spatial graphs.
method Generalization of Menasco's result to other classes of links, tangles, and spatial graphs.
result New more general results for tangles and spatial graphs.
Spatial graphs study tangle replacement with equivalence classes.
problem Differentiating spatial graphs and their properties.
method Tangle replacement on spatial graphs, focusing on handcuff graphs.
result One-to-one correspondence between neighborhood equivalence classes and tangles.
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.
We say that a graph is intrinsically non-trivial if every spatial embedding of the graph contains a non-trivial spatial subgraph. We prove that an intrinsically non-trivial graph is intrinsically linked, namely every spatial embedding of the graph contains a non-splittable 2-component link. We also show that there exis…
The paper classifies palettes of Dehn colorings for spatial graphs.
problem Classifying spatial graph diagrams using Dehn colorings.
method Examining vertex conditions and palettes for spatial graphs.
result Spatial graphs can be distinguished by the number of Dehn colorings with specific palettes.
Virtual knot theory is a generalization (discovered by the author in 1996) of knot theory to the study of all oriented Gauss codes. (Classical knot theory is a study of planar Gauss codes.) Graph theory studies non-planar graphs via graphical diagrams with virtual crossings. Virtual knot theory studies non-planar Gauss…
Improves graph convolutional networks by making their outputs smoother.
problem GCNs lack consideration for the smoothness of their output distributions against local perturbations.
method Introduces BVAT, a regularization method that generates virtual adversarial perturbations for graph-structured data.
result Establishes state-of-the-art results in semi-supervised node classification tasks.
We extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. To do this, we define the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. We prove that two graph grid diagrams representing the same tr…
Study on spatial graphs and their constituent knots, linking polynomial invariants.
problem Understanding the polynomial invariants of spatial graphs and their constituent knots.
method Analyzing spatial K4 graphs, constructing band surfaces, and relating polynomials. result Relations between Yamada/Jaeger polynomials and Jones polynomials of constituent knots and associated links.
We introduce an additional structure on ribbon graphs, arrow structure. We extend the Bollobás-Riordan polynomial to ribbon graph with this structure. The extended polynomial satisfies the contraction-deletion relations and naturally behaves with respect to the partial duality of ribbon graphs. We construct an arrow ri…
Extends knot concordance invariant to balanced spatial graphs using grid homology.
problem Defining a concordance invariant for balanced spatial graphs.
method Using grid homology to extend the invariant from knots to spatial graphs.
result The combinatorial Υ invariant is a concordance invariant for balanced spatial graphs. Study shows dense roots of Yamada polynomial for certain graphs.
problem Understanding the roots of Yamada polynomials for spatial graphs.
method Construction and analysis of Yamada polynomial for spatial graphs.
result Found an infinite family of graphs with dense roots of Yamada polynomials.
This article presents a survey of some recent results in the theory of spatial graphs. In particular, we highlight results related to intrinsic knotting and linking and results about symmetries of spatial graphs. In both cases we consider spatial graphs in S3 as well as in other 3-manifolds.
We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in …
Develops BASGCN for graph classification with improved feature learning.
problem Graph classification with information loss and imprecise representation.
method Transforms graphs into grid structures and defines a new spatial graph convolution operation.
result Reduces information loss and improves feature representation compared to existing models.
Classifies spatial graphs with finite N-quandles.
problem Determining isomorphism of spatial graphs' N-quandles.
method Generalized N-quandles to spatial graphs, proving basic results and conjecturing a classification.
result Verifies conjecture in several cases, presents a possible counterexample.
New formulas for spatial 2-bouquet graphs discovered.
problem Finding formulas for Vassiliev invariants of spatial 2-bouquet graphs.
method Introducing new Gauss diagram formulas for flat vertex isotopy classes of spatial 2-bouquet graphs.
result First simple example of a Gauss diagram formula for spatial 2-bouquet graphs.
A new method for stochastic optimization using virtual gradients.
problem Stochastic optimization challenges in computational efficiency and memory usage.
method Inspired by dynamic programming, SVGD uses a computational graph and automatic differentiation for efficient optimization.
result Experimental results show SVGD outperforms other methods on multiple datasets and network models.
The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.
problem Classifying virtual knot polynomials and trivalent graph invariants with specific conditions.
method Skein-theoretic techniques applied to classify invariants with smallness conditions.
result Classification of all non-trivial invariants of trivalent graphs and skein theories of virtual tangles.
In this paper, we show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.
We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy do…
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.