arXiv research

Characterizes curves with short representatives on hyperbolic surfaces.

We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, where the Riemannian metric $g$ and the magnetic field $σ$ are left-invariant. Our first result is that when $σ$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, then the associated…

In this note we study whether specific elements in the second homology of specific simply connected closed $4$-manifolds can be represented by smooth or topologically flat embedded spheres.

We measure, in two distinct ways, the extent to which the boundary region of moduli space contributes to the ``simple type'' condition of Donaldson theory. Using a geometric representative of μ(pt), the boundary region of moduli space contributes 6/64 of the homology required for simple type, regardless of the topology…

ENN neural network learns logical syllogisms using Euler diagrams.

There is an infinitely generated free subgroup of the smooth knot concordance group with the property that no nontrivial element in this subgroup can be represented by an alternating knot. This subgroup has the further property that every element is represented by a topologically slice knot.

We use an estimate on the Thurston--Bennequin invariant of a Legendrian link in terms of its Kauffman-polynomial to show that links of topological unknots, e.g. the Borromean rings or the Whithead link, may not be represented by Legendrian links of Legendrian unknots.

Extends Carathéodory's theorem to multidimensional domains with constant curvature.

New method integrates topological knowledge into data embeddings.

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…

We find numerical and empirical evidence for dynamical, structural and topological phase transitions on the (German) Frankfurt Stock Exchange (FSE) in the temporal vicinity of the worldwide financial crash. Using the Minimal Spanning Tree (MST) technique, a particularly useful canonical tool of the graph theory, two tr…

Proposes a probabilistic framework for smart contract risk quantification.

A major problem in the study of complex socioeconomic systems is represented by privacy issues$-$that can put severe limitations on the amount of accessible information, forcing to build models on the basis of incomplete knowledge. In this paper we investigate a novel method to reconstruct global topological properties…

Machine learning maps knots to embeddings, revealing topological invariants.

This research classifies singular foliations and finds a universal deformation.

The existence of topologically slice knots that are of infinite order in the knot concordance group followed from Freedman's work on topological surgery and Donaldson's gauge theoretic approach to 4-manifolds. Here, as an application of Ozsvath and Szabo's Heegaard-Floer theory, we show the existence of an infinite sub…

SM-netFusion estimates brain network atlas by considering multiple topological measures.

A new method learns node embeddings for signed directed networks by capturing both first-order and high-order topologies.

One of the main drawbacks of the practical use of neural networks is the long time required in the training process. Such a training process consists of an iterative change of parameters trying to minimize a loss function. These changes are driven by a dataset, which can be seen as a set of labelled points in an n-dime…

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold groupoid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent…

Robustly computes intrinsic coordinates on point clouds using resampling and averaging.

Novel theory combines combinatorial and topological elements.

Topology guidance controls generative model outputs by specifying topological features.

Neural nets learn robust geometric data representations.

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-deg…

Study index theory on Lie group homogeneous spaces using topological and analytic methods.

The subtle interplay between local and global charges for topological semimetals exactly parallels that for singular vector fields. Part of this story is the relationship between cohomological semimetal invariants, Euler structures, and ambiguities in the torsion of manifolds. Dually, a topological semimetal can be rep…

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples …

We study here global and local entanglements of open protein chains by implementing the concept of knotoids. Knotoids have been introduced in 2012 by Vladimir Turaev as a generalization of knots in 3-dimensional space. More precisely, knotoids are diagrams representing projections of open curves in 3D space, in contras…

This study optimizes cycle representatives in persistent homology using linear programming.

In \cite{Ka14} we produced an algorithm for deciding whether or not an element $φ\in Out(F_N)$ is an iwip ("fully irreducible") automorphism. At several points that algorithm was rather inefficient as it involved some general enumeration procedures as well as running several abstract processes in parallel. In this pape…

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…

Proposes a method to predict node attributes using network topology.

We introduce the concept of community trees that summarizes topological structures within a network. A community tree is a tree structure representing clique communities from the clique percolation method (CPM). The community tree also generates a persistent diagram. Community trees and persistent diagrams reveal topol…

Paper introduces TVaRD, a new topological risk measure for financial portfolios.

Graphoids are topological invariants of virtual graph diagrams.

Study shows certain toric arrangements have minimal topological complements.

In 1997, Collin proved that any properly embedded minimal surface in $\mathbb{R}^3$ with finite topology and more than one end has finite total Gaussian curvature. Hence, by an earlier result of Lopez and Ros, catenoids are the only non-planar, non-simply connected, properly embedded, minimal planar domains in $\mathbb…

Fractional Dehn twists give a measure of the difference between the relative isotopy class of a homeomorphism of a bordered surface and the Thurston representative of its free isotopy class. We show how to estimate and compute these invariants. We discuss the the relationship of our work to stabilization problems in cl…

Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.

In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give ev…

Topology-GS improves 3D GS for better structural and feature integrity.

This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.

Explains peculiarities of 4D scalar curvature via Yamabe invariant.

EuLearn creates diverse 3D topological datasets for machine learning.

Determining when two knots are equivalent (more precisely isotopic) is a fundamental problem in topology. Here we formulate this problem in terms of Predicate Calculus, using the formulation of knots in terms of braids and some basic topological results. Concretely, Knot theory is formulated in terms of a language with…

Topological method detects Hopf bifurcations from time series.

This paper introduces TDA and TSI for better business analytics.