Related papers: A statistical approach to knot confinement via per…
Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…
This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space…
Knots are entangled structures that cannot be untangled without a cut. Topological stability of knots is one of the many examples of their important properties that can be used in information storage and transfer. Knot dynamics is important…
Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or \v{C}ech filtration of $X$ to obtain what we call a persistence. This paper contains the…
By a fixed continuous map from a $3$-space to itself, a knot in the $3$-space may be mapped to another knot in the $3$-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze…
We present a computer simulation study of the compact self-avoiding loops as regards their length and topological state. We use a Hamiltonian closed path on the cubic-shaped segment of a 3D cubic lattice as a model of a compact polymer. The…
The interplay of geometrical and topological entanglement in semiflexible knotted polymer rings confined inside a spherical cavity is investigated using advanced numerical methods. By using stringent and robust algorithms for locating…
Long lived topological features are distinguished from short lived ones (considered as topological noise) in simplicial complexes constructed from complex networks. A new topological invariant, persistent homology, is determined and…
Hyperuniformity, the suppression of density fluctuations at large length scales, is observed across a wide variety of domains, from cosmology to condensed matter and biological systems. Although the standard definition of hyperuniformity…
We discuss and review recent developments in the area of applied algebraic topology, such as persistent homology and barcodes. In particular, we discuss how these are related to understanding more about manifold learning from random point…
We show that the average size of self-avoiding polygons (SAP) with a fixed knot is much larger than that of no topological constraint if the excluded volume is small and the number of segments is large. We call it topological swelling. We…
Topological constraints (TCs) between polymers determine the behaviour of complex fluids such as creams, oils and plastics. Most of the polymer solutions used every day life employ linear chains; their behaviour is accurately captured by…
We consider the folding of a self-avoiding homopolymer on a lattice, with saturating hydrogen bond interactions. Our goal is to numerically evaluate the statistical distribution of the topological genus of pseudoknotted configurations. The…
Detecting the dimension of a hidden manifold from a point sample has become an important problem in the current data-driven era. Indeed, estimating the shape dimension is often the first step in studying the processes or phenomena…
A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…
We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a…
We study several related models of self-avoiding polygons in a tubular subgraph of the simple cubic lattice, with a particular interest in the asymptotics of the knotting statistics. Polygons in a tube can be characterised by a finite…
An analysis of extensive simulations of interacting self-avoiding polygons on cubic lattice shows that the frequencies of different knots realized in a random, collapsed polymer ring decrease as a negative power of the ranking order, and…
We give two different, statistically consistent definitions of the length l of a prime knot tied into a polymer ring. In the good solvent regime the polymer is modelled by a self avoiding polygon of N steps on cubic lattice and l is the…
Persistent homology has been widely used to discover hidden topological structures in data across various applications, including music data. To apply persistent homology, a distance or metric must be defined between points in a point cloud…