Graph potentials link to topological QFTs, with computational methods.
problem Defining a topological quantum field theory using graph potentials.
method Using colored trivalent graphs and birational type to define a topological QFT.
result Graph potentials' birational type depends on the graph's homotopy type.
This paper proposes an axiomatic for Cyclic Foam Topological Field theories. That is Topological Field theories, corresponding to String theories, where particles are arbitrary graphs. World surfaces in this case are two-manifolds with one-dimensional singularities. We proved that Cyclic Foam Topological Field theories…
Polynomial algorithm found for alternating link equivalence.
problem Link equivalence of alternating links in 3-space.
method Tait flyping conjectures, observations from graph theory, and topological graph theory.
result Alternating link equivalence has a polynomial algorithm.
We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul an…
This paper refines understanding of decentralized learning by considering graph topology.
problem Current theory fails to predict performance in decentralized learning settings.
method Quantifies how graph topology influences convergence in decentralized learning.
result Graph topology significantly impacts convergence in decentralized learning, contrary to spectral gap theory.
New technique connects graph matching complexes to Morse theory for better topology understanding.
problem Understanding the topology of matching complexes of complete graphs.
method Developed discrete Morse theory technique to analyze Mn. result Showed Mn is geometrically (νn−1)-connected, improving on previous homotopical results. In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…
New TQFT homologies help color graphs, potentially solving the four color theorem.
problem Graph coloring problem, especially the four color theorem.
method Topological quantum field theory (TQFT) to define homology theories.
result TQFT homologies can generate 4-face colorings of bridgeless planar graphs, offering a constructive approach to the four color theorem.
Classifies graph configuration spaces homeomorphic to manifolds.
problem Classifying graph configuration spaces homeomorphic to manifolds.
method Developed techniques to translate topological properties into graph theoretic ones.
result Extended Abrams' work to classify certain graph configuration spaces.
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory, combinatorics, and singularity theory. The multidimensional analogues of this complex ar…
Paper categorifies a polynomial related to ribbon graphs.
problem Enumerating partial duals of ribbon graphs.
method Using an extended Frobenius algebra in unoriented topological quantum field theory.
result A categorification of the partial-dual genus polynomial.
Graph coloring is explained using a topological field theory with defects.
problem Graph coloring as a combinatorial problem is quantum in nature.
method Topological field theory with defects to interpret graph coloring.
result Graph coloring is related to sections of a certain bundle.
We first show that the braid group over a graph topologically containing no Θ-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a r…
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.
This work introduces a method to compare sparse neural network topologies using graph theory.
problem Comparing and understanding sparse neural network topologies, especially during training.
method Introducing Neural Network Sparse Topology Distance (NNSTD) to measure distances between different sparse neural networks.
result Sparse neural networks can outperform over-parameterized models without further structure optimization.
Well-quasi-orders proved on embedded planar graphs.
problem Proving well-quasi-orders on embedded planar graphs.
method Careful analysis and extensions of classical methods for embedded minor relations.
result Embedded minor relations are well-quasi-orders on various classes of embedded planar 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…
Quantum model for knotted graphs from knot theory.
problem Constructing an isotopy invariant polynomial for knotted bipartite ribbon graphs.
method Applying quantum topology to construct an isotopy invariant polynomial.
result Computed the expected number of loops in the double dimer model.
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.
Paper describes a state sum formula for a graph coloring polynomial.
problem Counting n-face colorings of ribbon graphs for various n. method Combines topological quantum field theory and diagrammatic tensors.
result Describes a state sum formula for the total face color polynomial.
We extend the topological field theory (``itsy bitsy topological field theory"') of our previous work from mod-2 to twisted coefficients. This topological field theory is derived from sutured Floer homology but described purely in terms of surfaces with signed points on their boundary (occupied surfaces) and curves on …
One of the main questions in the theory of normal surface singularities is to understand the relations between their geometry and topology. The lattice cohomology is an important tool in the study of topological properties of a plumbed 3-manifold M associated with a connected negative definite plumbing graph G. It conn…
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…
Graph manifolds with small homology have non-trivial SU(2) representations.
problem Characterizing graph manifolds with small homology.
method Topological methods avoiding gauge theory.
result Existence of irreducible SU(2) representations for certain graph manifolds.
Paper presents voxel graph operators for vector data models.
problem Efficient conversion and analysis of geometric models.
method Topological voxelization, graph construction, differential operator derivation.
result Discrete differential and integral operators from voxel complexes.
PiNGDA learns beneficial noise for graph augmentation stability.
problem Challenges in generating effective and stable graph augmentations.
method PiNGDA uses positive-incentive noise to scientifically analyze and generate beneficial graph augmentations.
result PiNGDA improves GCL performance by learning beneficial noise on graph topology and attributes.
AdaCGP learns dynamic graph topology from time series data, improving over existing methods.
problem Learning dynamic graph topology from time-varying signals, especially in real-time applications.
method AdaCGP is a sparsity-aware adaptive algorithm that recursively estimates the Graph Shift Operator (GSO) through variable splitting.
result AdaCGP outperforms state-of-the-art methods in GSO estimation, achieving improvements exceeding 83%.
The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in th…
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics.…
Proposes a probabilistic framework for stationary topological signals on simplicial complexes.
problem Complex data structures require new models and tools.
method Generalizes stationarity to topological signals on simplicial complexes.
result Defines topological power spectral density (PSD) for stationary signals.
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered n-color vertex homology for n=2 is generated by perfect matchings. Consider a finite, regular cover Y→X of finite graphs, with associated deck group G. We relate the topology of the cover to the structure of H1(Y;C) as a G-representation. A central object in this study is the {\em primitive homology} group $H_1^{\mathrm{prim}}(Y;\mathbb{C})\subseteq H_1(Y;\mathbb{…
Extends knotoid theory to include multiple poles and intervals.
problem No new problem introduced.
method Definition of generalized knotoids and graphs, exploration of invariants.
result Theory subsumes various topological objects and introduces new cases.
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
problem Classifying steady Euler flows with Morse-Bott Bernoulli functions.
method Constructing non-vanishing steady solutions using integrable systems and topology.
result Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph three-manifolds.
New topological realization of Kontsevich graph complex for large dimensions.
problem Understanding the rational homotopy groups of Diff partial(D2k).
method Construction of a chain map from Kontsevich graph complex to rational singular chain complex.
result New elements in rational homotopy groups of BDiff partial(D2k) determined by cycles in graph complex.
The curve graph's model theory reveals its central role in surface study.
problem Why is the curve graph central in surface and mapping class group studies?
method Developed a bridge between model theory, topology, and group theory; bi-interpreted curve graph with mapping class group.
result Proved the curve graph's first-order theory is ω-stable and has quantifier elimination. Solved Cheeger inequalities for simplicial complexes, combining topological and graph theoretic methods.
problem Extend Cheeger inequalities to simplicial complexes and their higher order Laplacians.
method Combining constructions from simplicial topology, signed graphs, Gromov filling radii, and interpolating between 1-Laplacians and 2-Laplacians.
result Developed a general theory for p-Laplacians on simplicial complexes and proved Cheeger-type inequalities.
New framework for 3D spatial topology enumeration and identification.
problem Efficient navigation through complex engineering system topologies.
method Mathematical spatial graph theory to represent, enumerate, and identify unique topological classes.
result Identification of distinctive 3D topological classes for engineering systems.
Non-relatively hyperbolic separating curve graph for surfaces.
problem Classifying hyperbolicity of separating curve graphs.
method Proof of non-relatively hyperbolic property.
result Separating curve graph is not relatively hyperbolic for surfaces with genus ≥ 3 and one boundary component.
Researchers identify graph components for unicellular collections.
problem Understanding connected components of surgery graph for unicellular collections.
method Group-theoretic approach involving mapping class group action.
result Connected components enumerated by a homological invariant.
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.
New model learns from random graph samples to estimate graph parameters.
problem Scalability issues in graph learning methods for large graphs.
method Develops a graph classification model working on randomly sampled subgraphs.
result Validates mini-batch learning on graphs and provides generalization bounds.
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…
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.
This paper develops a discrete theory of real Riemann surfaces using quad-graphs and linear discretization.
problem Constructing a discrete theory of real Riemann surfaces.
method Using quad-graphs and linear discretization of Cauchy-Riemann equations, constructing a symplectic homology basis.
result The discrete period matrix has the same canonical decomposition as in the smooth setting.
Lecture notes on crystallography and discrete surfaces.
problem Mathematical modeling of crystal structures.
method Variational principle and discrete surface theory.
result Most symmetric crystal structures identified.
The paper explores invariants of graph drawings in the plane.
problem Understanding the invariants of almost embeddings of graphs in the plane.
method Proves relations between invariants, connects to homology, constructs examples.
result Constructs almost embeddings realizing some values of invariants.