Homology and cohomology theory for topological quandles computed.
problem Computing invariants for knot diagrams using quandle cocycles.
method Introducing homology and cohomology theory for topological quandles, studying their relation to quandle groups, and using topological quandle cocycles to compute state sum invariants.
result State sum invariants computed using topological quandle cocycles.
A new method for state estimation on complex networks.
problem Reconstructing latent dynamics from multivariate time-series on topological cell complexes.
method Topology-aware state space framework derived from stochastic partial differential equations, with state evolution following heat-like topological diffusion.
result The proposed method successfully recovers latent states and topological structures in real-world networks.
Constructs a path integral for fermionic SPTs, solving anomalies in 2+1D topological orders.
problem Anomalies in (2+1)D fermionic topological phases and their computation.
method Combining (2+1)D fermionic topological order with symmetry fractionalization data to construct a (3+1)D path integral.
result Reproduces the Z16 anomaly indicator for time-reversal symmetric topological superconductors. This study uses persistent homology to analyze complex transitional networks from time series data.
problem Lack of effective tools to summarize complex topology in transitional networks.
method Persistent homology from topological data analysis applied to coarse-grained state-space networks (CGSSN).
result CGSSN improves dynamic state detection and noise robustness compared to other methods.
Introduces sheaf quantization, a topological approach to geometric quantization.
problem Topological realization of WKB-states in geometric quantization.
method Enhancement of constructible sheaves, Betti counterpart of Fukaya--Floer theory.
result Introduction to sheaf quantization as a topological realization of WKB-states.
This paper formalizes state similarity metrics for reinforcement learning.
problem Leveraging state similarity for reinforcement learning in continuous-state systems.
method Introducing a unified formalism for defining topologies through metrics.
result Established a hierarchy of metrics and demonstrated their theoretical implications.
Contagions such as the spread of popular news stories, or infectious diseases, propagate in cascades over dynamic networks with unobservable topologies. However, "social signals" such as product purchase time, or blog entry timestamps are measurable, and implicitly depend on the underlying topology, making it possible …
Method learns topological states from randomized measurements.
problem Detecting topologically ordered two-dimensional states on quantum processors.
method Variational tensor network tomography with randomized measurements.
result Demonstrated ability to learn ground states of surface code and quantum spin liquid states.
Fixed point assertions in digital topology are often incorrect or poorly stated.
problem Fixed points in digital metric spaces
method Discussing publications with bad assertions
result Identifying and correcting errors in fixed point assertions
We demonstrate the triangulability of compact 3-dimensional topological pseudomanifolds and study the properties of such triangulations, including the Hauptvermutung and relations by Alexander star moves and Pachner bistellar moves. We also provide an application to state-sum invariants of 3-dimensional topological pse…
The main approaches for node classification in graphs are information propagation and the association of the class of the node with external information. State of the art methods merge these approaches through Graph Convolutional Networks. We here use the association of topological features of the nodes with their clas…
Paper speeds up topological signal identification and cycle matching.
problem Efficiently identifying and matching topological signals across datasets.
method Cohomological approach to persistent homology computation.
result Significantly faster performance on large-scale datasets.
A novel topological method analyzes fMRI data over time.
problem Analyzing time-varying fMRI data due to noise and person-to-person variation.
method Encoding each time point as a persistence diagram of topological features.
result Time-varying persistence diagrams can cluster participants and study brain state trajectories.
New method detects and compares folding pathways of knotted proteins.
problem Understanding the function of knots in protein folding.
method Topological analysis of protein knotoid distributions and entanglement.
result Reveals unique folding pathway for shallow knotted Carbonic Anhydrases.
Topological theory for qLDPC codes enables non-Clifford gates and magic state injection.
problem Fault-tolerant quantum computation in qLDPC codes with non-Clifford gates and magic state resources.
method Developed a topological theory using simplicial or CW complex structures and deformation retraction.
result Achieved non-Clifford gates and magic state injection in qLDPC codes with constant rate and polynomial distance.
3-manifolds study Hasse norm principle, akin to number fields.
problem Topological analog of Hasse norm principle for 3-manifolds.
method Analogy with number fields and finite cyclic coverings.
result Showed topological Hasse norm principle for 3-manifolds.
Gerbes encode spectral gaps in topological insulators.
problem Capturing spectral gaps in topological insulators.
method Geometric encoding through 'Real' gerbes.
result Gerbes precisely capture spectral gaps.
Proposes a new method for disentangling data representations using topological analysis.
problem Learning disentangled representations for better model explainability and robustness.
method Integrates a multi-scale topological loss term into the training of deep learning models.
result Improves disentanglement scores compared to state-of-the-art methods.
Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
In this paper we state and prove the analogous of the principal ideal theorem of algebraic number theory for the case of 3-manifolds from the point of view of arithmetic topology.
Proposes a constraint for deep clustering to handle both simple and complex topologies.
problem Limited prior knowledge for deep clustering methods to perform well on complex topologies.
method Introduces a constraint using symmetric InfoNCE to enhance deep clustering performance.
result The constraint improves deep clustering methods' performance on both simple and complex topologies.
We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group Oq2(SL(2)) providing a topological interpretation for its structure morphisms. We also show that its sta…
The paper characterizes boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
problem Characterizing boundaries in Turaev-Viro TQFTs and Dijkgraaf-Witten theories.
method Identifying explicit boundary locality conditions and proving consistency with state sum models.
result Turaev-Viro and Dijkgraaf-Witten theories with boundary defects admit a state sum description.
In this paper we give a characterization of 2-dimensional topological field theories over a space X as Frobenius bundles with connections over LX, the free loop space of X. This is a generalization of the folk theorem stating that 2-dimensional topological field theories (over a point) are described by finite-dim…
Study geometric flows with varying parameters and prove continuous dependence.
problem Continuous dependence of flows on parameters in geometric settings.
method Derived suitable topologies for vector fields and flows, proved new continuous dependence.
result Proved continuous dependence of flows on parameters in a general topological space.
New pseudo-Hermitian models from non-semisimple TQFTs.
problem Constructing exactly solvable pseudo-Hermitian spin Hamiltonians.
method Identifying ground states on surfaces using non-semisimple TQFTs.
result Ground states depend only on spatial topology and can be assigned by non-semisimple TQFTs.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect or poorly stated fixed point assertions in digital topology.
method Discussion of problematic publications in digital metric spaces.
result Clarification of incorrect fixed point assertions.
Incorrect fixed point assertions in digital topology are discussed.
problem Incorrect, incorrectly proven, or trivial fixed point assertions in digital topology.
method Continues earlier work on identifying and critiquing bad fixed point assertions.
result Clarifies the nature and extent of incorrect fixed point assertions in digital topology.
We study the crash dynamics of the Warsaw Stock Exchange (WSE) by using the Minimal Spanning Tree (MST) networks. We find the transition of the complex network during its evolution from a (hierarchical) power law MST network, representing the stable state of WSE before the recent worldwide financial crash, to a superst…
Graph neural network using Beltrami flow for feature and topology evolution.
problem Efficient feature learning and topology evolution on graphs.
method Discretized Beltrami flow applied to graph neural networks with positional encodings.
result Achieves state-of-the-art results on various benchmarks.
Proves a conjecture about graph complexes without specific cycle lengths.
problem Graph complexes without specific cycle lengths.
method Proves stronger statements about independence complexes being contractible or homotopy equivalent to spheres.
result Independence complexes are either contractible or homotopy equivalent to spheres.
We use a semisupervised learning algorithm based on a topological data analysis approach to assign functional categories to yeast proteins using similarity graphs. This new approach to analyzing biological networks yields results that are as good as or better than state of the art existing approaches.
We extend the well-known Borromean and Brunnian rings to new higher order versions. Then we suggest an extension of the connection between Efimov states in cold gases and Borromean and Brunnian rings to these new higher order links. This gives rise to a whole new hierarchy of possible states with Efimov states at the b…
We explore 4d Yang-Mills gauge theories (YM) living as boundary conditions of 5d gapped short/long-range entangled (SRE/LRE) topological states. Specifically, we explore 4d time-reversal symmetric pure YM of an SU(2) gauge group with a second-Chern-class topological term at θ=π (SU(2)θ=π YM), by turning on backg…
This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies employed in the proofs. Where appropriate we give statements for manifolds of all di…
New method detects dangerous states in gridworlds using geometric defects.
problem Detecting dangerous states in gridworlds for AI safety.
method Modified state complexes and Gromov's Link Condition.
result Geometric defects indicate dangerous states in gridworlds.
TopoFisher learns topological summaries by maximizing Fisher information, improving parameter efficiency and inference quality.
problem Simulation-based inference misses key information in low-order statistics, especially for non-Gaussian fields.
method TopoFisher uses a differentiable persistent-homology pipeline that learns topological summaries by maximizing local Gaussian Fisher information.
result TopoFisher recovers much of the available information and outperforms fixed topological vectorizations in weak gravitational lensing.
In this note we provide natural optimal geometric conditions for a Riemannian manifold suitably covered by two open metric balls to be homeomorphic to a sphere. This can be viewed as a geometric analogue of Brown's theorem in topology stating that a closed manifold covered by two topological balls is a sphere.
We conjecture a relation between the sl(N) knot homology, recently introduced by Khovanov and Rozansky, and the spectrum of BPS states captured by open topological strings. This conjecture leads to new regularities among the sl(N) knot homology groups and suggests that they can be interpreted directly in topological st…
Graph neural controlled differential equations learn graph dynamics from vertex observations.
problem Predicting future states of dynamical systems on graphs with limited vertex data.
method Incorporates graph topology information into NCDE to predict graph dynamics.
result Informed NCDE requires fewer parameters and lower MAE compared to previous methods.
We derive the general state sum construction for 2D topological quantum field theories (TQFTs) with source defects on oriented curves, extending the state-sum construction from special symmetric Frobenius algebra for 2-D TQFTs without defects (cf. Lauda \& Pfeiffer \cite{LP}). From the extended Pachner moves (Crane \& …
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 introduce the notion of a "state function" for framed tangles in a disk. After choosing a finite set of states for each marked disk, a state function is a projection from the vector space spanned by all tangles to the vector space spanned by the states, that is local, and topologically invariant. Given the states fo…
In this paper we discuss algebraic, combinatorial and topological properties of singular virtual braids. On the algebraic side we state the relations between classical and virtual singular objects, in addition we discuss a Birman-like conjecture for the virtual case. On the topological and combinatorial side, we prove …
We introduce a Bayesian approach to discovering patterns in structurally complex processes. The proposed method of Bayesian Structural Inference (BSI) relies on a set of candidate unifilar HMM (uHMM) topologies for inference of process structure from a data series. We employ a recently developed exact enumeration of to…
Power series invariant of hyperbolic 3-manifolds matches knot invariants.
problem Understanding topological invariants of hyperbolic 3-manifolds.
method Perturbative power series associated with ideally triangulated cusped hyperbolic 3-manifolds.
result The power series agrees with Kashaev and Andersen-Kashaev invariants to all orders.
Efficiently sparsifies simplicial complexes using local densities of states.
problem Prohibitive computational requirements for dense simplicial complexes.
method Probabilistic sparsification using local densities of states and kernel-ignoring decomposition.
result Approximates the spectrum of the original SC with a sparser surrogate SC.
This work analyzes PPR-based node embeddings and their topological information.
problem Understanding and interpreting PPR-based node embeddings.
method Unified framework and two methods for topology recovery.
result PPR-based embeddings maintain more topological information than random walk-based embeddings.