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.
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.
GALA framework learns invariant graph representations via environment augmentation with minimal assumptions.
problem Learning invariant graph representations from different environments without additional assumptions.
method Developed GALA framework with minimal assumptions of variation sufficiency and consistency. Uses an assistant model to differentiate graph environment changes.
result Extracting maximally invariant subgraphs to proxy predictions identifies underlying invariant subgraphs for successful out-of-distribution generalization.
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. We introduce invariants of spatial graphs related to the Wu invariant and the Simon invariant, and apply them to prove that certain graphs are intrinsically chiral, and to obtain lower bounds for the minimal crossing number of embedded graphs.
Graph homomorphism numbers embed graphs for classification.
problem Graph classification using graph homomorphisms.
method Embed graphs into vectors using homomorphism numbers.
result Homomorphism vectors are universal for approximating graph invariants.
We define the fundamental quandle of a spatial graph and several invariants derived from it. In the category of graph tangles, we define an invariant based on the walks in the graph and cocycles from nonabelian quandle cohomology.
Novel framework improves graph learning for out-of-distribution generalization.
problem Graph out-of-distribution generalization challenges in neural networks.
method Invariant Graph Learning based on Information bottleneck theory (InfoIGL).
result Achieves state-of-the-art performance in graph classification tasks under OOD generalization.
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.
This is a short review article on invariants of spatial graphs, written for "A Concise Encyclopedia of Knot Theory" (ed. Adams et. al.). The emphasis is on combinatorial and polynomial invariants of spatial graphs, including the Alexander polynomial, the fundamental quandle of a graph, and the Yamada polynomial.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
New neural architectures invariant to sign flips and basis symmetries for graph representation learning.
problem Learning invariant graph representations from eigenvectors.
method SignNet and BasisNet neural architectures that are invariant to sign flips and basis symmetries.
result Proven to be universal, approximating any continuous function of eigenvectors with desired invariances.
This paper proves a conjecture linking quantum modular forms and WRT invariants for specific graphs.
problem Proving a conjecture about quantum modular forms and WRT invariants for unimodular H-graphs.
method Constructed finite sums of rational functions, studied weighted Gauss sums, and combined results to prove the conjecture.
result WRT invariants of H-graphs yield quantum modular forms of depth two and weight one.
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.
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.
Generative models of graph structure have applications in biology and social sciences. The state of the art is GraphRNN, which decomposes the graph generation process into a series of sequential steps. While effective for modest sizes, it loses its permutation invariance for larger graphs. Instead, we present a permuta…
The paper explores weight systems and their applications to graph and embedded graph invariants.
problem Developing weight systems for graphs and embedded graphs.
method Construction of weight systems from graph invariants and metrized Lie algebras, and extending to arbitrary embedded graphs.
result Explicit forms of generating functions and recurrence relations for weight systems on chord diagrams and embedded graphs.
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 sl(n) polynomial to knotted 4-valent graphs. Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. Fleming and the author introduced some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the constituent 2-component algebraically split li…
Learning generative models for graph-structured data is challenging because graphs are discrete, combinatorial, and the underlying data distribution is invariant to the ordering of nodes. However, most of the existing generative models for graphs are not invariant to the chosen ordering, which might lead to an undesira…
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.
We define some signature invariants for a class of knotted trivalent graphs using branched covers. We relate them to classical signatures of knots and links. Finally, we explain how to compute these invariants through the example of Kinoshita's knotted theta graph.
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.
Defines a new invariant from graph configurations in three-manifolds.
problem Counting embeddings of graphs in rational homology spheres.
method Uses integrals on configuration spaces of points in the manifold.
result Defines the degree two part of the logarithm of the invariant for concrete computations.
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 …
We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of Uq(sl2). We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…
kth-order invariant graph networks are as powerful as kth-order WL in distinguishing graphs.
problem Measuring the expressive power of graph neural network formalisms.
method Considered kth-order invariant graph networks (k-IGNs) and compared their expressive power to kth-order WL.
result k-IGNs and k-WL are equally powerful in distinguishing graphs.
Paper introduces new link homotopy invariants and applies them to 3-bouquet graphs.
problem Milnor's triple linking number and its applications in link homotopy.
method Developed new integer-valued link homotopy invariants and applied them to 3-bouquet graphs.
result Found new integer-valued invariants derived from four terms summing to Milnor's triple linking number.
New knot invariant from 3-braids and 6-valent graphs.
problem Classical knot invariant construction.
method Using group Gn3 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 consider the problem of undirected graphical model inference. In many applications, instead of perfectly recovering the unknown graph structure, a more realistic goal is to infer some graph invariants (e.g., the maximum degree, the number of connected subgraphs, the number of isolated nodes). In this paper, we propo…
Constructs a 4-invariant for graphs at c = 3/8.
problem No specific problem stated; focuses on construction.
method Constructs a 4-invariant that extends a specialization of the sl(2)-weight system at c = 3/8, satisfying a deletion-contraction relation.
result Satisfies a simple deletion-contraction relation.
The paper defines invariants for almost graph embeddings and explores their properties.
problem Understanding the properties and limitations of almost graph embeddings in the plane.
method Introducing and analyzing integer invariants (winding number, Wu numbers) for almost embeddings.
result Some values of invariants are realizable for almost embeddings but not for embeddings.
Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnor's link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples…
We describe two locally finite graphs naturally associated to each knot type K, called Reidemeister graphs. We determine several local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring. Lastly, we introduce another object, relating t…
In this paper we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many bounda…
The study provides a criterion to compute the total Thurston-Bennequin invariant of Legendrian graphs.
problem Computing the total Thurston-Bennequin invariant for Legendrian graphs.
method Generalized criterion for computing the total Thurston-Bennequin invariant from the tb of smaller cycles.
result The criterion holds for graphs with up to 9 vertices and for infinite families of examples.
It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a mar…
Quotients of Gordian and H(2)-Gordian graphs are hyperbolic.
problem Investigate quotients of Gordian and H(2)-Gordian graphs under knot invariants.
method Defined equivalence relations by knot invariants (det, Jones span, tricolorability) and showed quotient graphs are Gromov hyperbolic.
result Quotients of H(2)-Gordian graph of links modulo span of Jones polynomial is isomorphic to complete graph.
New invariant defined for Weinstein domains, related to Kirby-Thompson's invariant.
problem Defining a symplectic invariant for Weinstein domains.
method Contact cut graph, Lefschetz fibrations, multisections with divides.
result Definition of Weinstein L-invariant and its relation to Kirby-Thompson's invariant. New invariant for special alternating links based on graph Laplacian.
problem Developing an invariant for special alternating links.
method Using the Laplacian matrix of the Tait graph, invariant is defined.
result A specific quadratic trace expression is invariant under flype moves.
IsoGCNs learn invariant and equivariant graph features for efficient simulations.
problem Learning isometric transformation invariant and equivariant features in graphs for simulations.
method Transformation invariant and equivariant Graph Convolutional Networks (IsoGCNs).
result IsoGCNs outperform state-of-the-art methods on geometrical and physical simulation tasks.
New invariant counts graph configurations in 3D manifolds.
problem Counting graph configurations in 3D manifolds.
method Using combings instead of parallelizations for a more flexible definition.
result Universal finite type invariant of three-manifolds.
Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular tra…
The paper extends a knot invariant to graphs and connects it to homology cylinders.
problem Understanding the structure of homology cobordism groups.
method Using tangle Floer homology, the authors define a new invariant for embedded graphs and prove a concatenation formula.
result The new invariant induces a homomorphism on the homology cobordism group of homology cylinders.
There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handl…
Refines Ozsváth-Szabó d-invariants for knot concurrence.
problem Computing Ozsváth-Szabó d-invariants for specific knot types.
method Refines and applies Karakurt and Şavk's formula for surgeries on almost simple linear graphs.
result Inequalities for d-invariants are equalities or strict in specific families.
Study shows surfaces without certain curves have infinite orbit graph.
problem Characterizing surfaces with specific curve properties.
method Utilized tools from mapping class group geometry.
result Infinite-invariance index 1 surfaces lack good curve graphs.
Geometric deep learning predicts knot invariants.
problem Predicting knot invariants from knot data.
method Constructing a functor from knots to graphs and using graph neural networks.
result High generalization capabilities demonstrated.