arXiv research

New method reduces spatial graphs while preserving their topological features.

New framework for 3D spatial topology enumeration and identification.

Graphoids are topological invariants of virtual graph diagrams.

Algorithm determines spatial graph isomorphism with vertex, edge colorings and orientations.

This is a survey article for the forthcoming `A Concise Encyclopedia of Knot Theory.' We focus on the topology of spatial graphs with few vertices and edges, paying particular attention to Brunnian $θ$-graphs.

Classical knot theory can be generalized to virtual knot theory and spatial graph theory. In 2007, Fleming and Mellor combined virtual knot theory and spatial graph theory to form, combinatorially, virtual spatial graph theory. In this paper, we introduce a topological definition of virtual spatial graphs that is simil…

This paper identifies all topological symmetry groups for Heawood family graphs.

We use the topological invariant of spatial graphs introduced by S. Yamada to find necessary conditions for a spatial graph to be periodic with a prime period. The proof of the main result is based on computing the Yamada skein algebra of the solid torus then proving that this algebra injects into the Kauffman bracket …

Proposes a method to forecast spatial-temporal data with limited training data.

We give a necessary and sufficient condition for the mapping class group of the pair of the 3-sphere and a graph embedded in it to be isomorphic to the topological symmetry group of the embedded graph.

Extends knotoid theory to include multiple poles and intervals.

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…

Paper presents voxel graph operators for vector data models.

Spatial and time-dependent data is of interest in many applications. This task is difficult due to its complex spatial dependency, long-range temporal dependency, data non-stationarity, and data heterogeneity. To address these challenges, we propose Forecaster, a graph Transformer architecture. Specifically, we start b…

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…

This paper proposes a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model that can directly learn a classification function for graphs of arbitrary sizes. Unlike state-of-the-art Graph Convolutional Neural Network (GCNN) models, the proposed QSGCNN model incorporates the process of identifying transit…

The spatial convolution layer which is widely used in the Graph Neural Networks (GNNs) aggregates the feature vector of each node with the feature vectors of its neighboring nodes. The GNN is not aware of the locations of the nodes in the global structure of the graph and when the local structures corresponding to diff…

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in $S^3$.

Proposes deep graph persistence to address neural persistence issues in deep learning.

In 1983, Conway-Gordon showed that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruen…

Graph neural controlled differential equations learn graph dynamics from vertex observations.

New method finds knots without low treewidth diagrams.

Accurate and real-time traffic forecasting plays an important role in the Intelligent Traffic System and is of great significance for urban traffic planning, traffic management, and traffic control. However, traffic forecasting has always been considered an open scientific issue, owing to the constraints of urban road …

A3T-GCN improves traffic forecasting by capturing spatial and temporal dependencies.

Proves minimal crossing diagrams for specific spatial graphs.

New invariants distinguish spatial graphs not previously possible.

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

Grid homology theory for spatial graphs extends skein sequence.

Defines concordance for spatial graphs and proves sliceness equivalence.

New results on splitting tangles and spatial graphs.

Spatial graphs study tangle replacement with equivalence classes.

Spatial graphs are decomposed into planar forests and braids.

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.

Modern intelligent transportation systems provide data that allow real-time dynamic demand prediction, which is essential for planning and operations. The main challenge of prediction of dynamic Origin-Destination (O-D) demand matrices is that demands cannot be directly measured by traffic sensors; instead, they have t…

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.

Extends knot concordance invariant to balanced spatial graphs using grid homology.

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 $S^3$ 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 …

New formulas for spatial 2-bouquet graphs discovered.

Classifies spatial graphs with finite N-quandles.

Graph convolutional networks (GCNs) are powerful tools for graph-structured data. However, they have been recently shown to be vulnerable to topological attacks. To enhance adversarial robustness, we go beyond spectral graph theory to robust graph theory. By challenging the classical graph Laplacian, we propose a new c…

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…

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.

Complete classification of links and spatial graphs with finite N-quandles.

Paper generalizes pretzel links using spatial graphs.

For leveled spatial graphs, we find a surface embedding that allows cellular embedding.