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.
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.
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.
New model preserves graph structure in large datasets.
problem Lack of permutation invariance in graph generation models for large graphs.
method Uses graph embeddings to create a scalable generative model.
result Model maintains structure in large graphs without losing invariance.
A new invariant for knotted graphs defined by label bracket.
problem Defining an invariant for knotted trivalent graphs.
method Generalizing Akimova and Manturov's construction to define the label bracket.
result The label bracket defines an isotopy invariant of knotted trivalent graphs.
A new method learns graph distributions invariant to node ordering.
problem Graphs are hard to model due to node ordering invariance issues.
method Score-based generative modeling with permutation equivariant graph neural network.
result The method achieves better or comparable graph generation results.
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.
Graphs from knot types help identify unique knots.
problem Identifying knots uniquely.
method Created Reidemeister graphs from knot types and analyzed their properties.
result Graph isomorphism type is a complete knot invariant.
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.
New method for testing graph invariants under undirected models.
problem Testing graph invariants under undirected models.
method Skip-down algorithm for monotone graph invariants.
result Optimal and adaptive confidence intervals for graph invariants.
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 invariants detect a specific graph in spatial webs.
problem Detecting specific graphs in spatial webs.
method Introduced new invariants and used spectral sequences.
result Proved invariants detect the planar theta graph.
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.
Characterizes invariant and equivariant linear layers for graphs.
problem Maximal collection of invariant and equivariant linear layers for graphs.
method Characterization of all permutation invariant and equivariant linear layers for graphs.
result Dimension of linear layers for edge-value graph data is 2 and for k-tuples of nodes, it is the k-th and 2k-th Bell numbers.
New invariant shows 4D graph-manifolds' covers match orthogonal types.
problem Understanding covers of 4D graph-manifolds.
method Introduced a topological invariant to prove universal covers' equivalence.
result 4D graph-manifolds' universal covers are bi-Lipschitz equivalent to orthogonal graph-manifold covers.
PiNet learns graph representations invariant to node permutations.
problem Graph classification and representation learning invariant to node permutations.
method Differentiable node attention pooling, permutation invariant graph neural network.
result Significant accuracy improvement in isomorphic graph classification with limited training data.
New graph-based invariants from quandle cocycles.
problem Defining new link invariants from quandle cocycles.
method Integrating quandle cocycle information into quandle coloring quivers to create weighted directed graphs.
result Definition of new link invariants including a 2-variable polynomial.
New τ invariant for balanced spatial graphs, extending knot concordance invariant.
problem Defining a concordance invariant for spatial graphs.
method Combinatorial definition and Heegaard Floer homology theory.
result Well-defined τ invariant for balanced spatial graphs and links.
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.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
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.
New invariant for surface-knots in 4D from marked graphs.
problem Invariants for surface-knots in 4D.
method Marked graph diagrams for surface-knots.
result Invariant constructed for surface-knots.
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…
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.
Alexander polynomial defined for MOY graphs.
problem Defining an Alexander polynomial for MOY graphs.
method Introduced a refined Alexander polynomial for framed trivalent MOY graphs.
result Invariant satisfies MOY-type relations and defines Alexander polynomial of links.
Defines Hopf monoid of directed graphs and its invariant.
problem Defining a Hopf monoid for directed graphs.
method Defines Hopf monoid of directed graphs and shows embedding in GP.
result Invariant of directed graphs coincides with strict chromatic polynomial.
New invariants for Legendrian graphs defined and proven.
problem Defining and proving invariants for Legendrian graphs.
method Defined ruling invariants and proved their properties.
result Ruling invariants are compatible with vertex-identifying operations and vertical cuts/gluings.
Categorifies invariants of 3-manifolds using handlebody graphs.
problem Invariants of 3-manifolds from different categories.
method Graphical calculus on handlebodies, semi-simple and spherical categories.
result Recover Kuperberg and Turaev-Viro 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.
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 …
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.
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…
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.
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.
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.
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…
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.