Curvature of random walks and random polygons in confinement

Diao, Yuanan; Ernst, Claus; Montemayor, Anthony; Ziegler, Uta

doi:10.1088/1751-8113/46/28/285201

Cited by 5 publications

(8 citation statements)

References 19 publications

(41 reference statements)

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“…Interestingly, unlike in the case of the ACN, we do not find any significant correlation between either the curvature or torsion of the knots and the integral I. This should be compared with other more ad hoc approaches, such as [34][35][36].…”

Section: Resultscontrasting

confidence: 53%

A statistical approach to knot confinement via persistent homology

Celoria

Mahler

2022

Proc. R. Soc. A.

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In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot’s embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris–Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots’ lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots.

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Section: Resultscontrasting

confidence: 53%

A statistical approach to knot confinement via persistent homology

Celoria

Mahler

2022

Proc. R. Soc. A.

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“…In fact, numerical observations based on long confined random walks led some of the authors to the following conjecture [12]: as L → ∞ and R → 1/2, the average torsion (per edge) of a random polygon in confinement approaches π/3.…”

Section: Total Curvature and Total Torsion For Confined Phantom Polygonsmentioning

confidence: 99%

“…The effects of knot complexity and length on total curvature and total torsion have been studied before for unconfined random knotted polygons for a few simple knot types [28]. In [12] the authors derive an expression for the total curvature of confined equilateral random walks (i.e. confined open chains) that is independent of knotting and compare this result with numerical data obtained from confined equilateral random polygons.…”

Section: Introductionmentioning

confidence: 99%

Total curvature and total torsion of knotted random polygons in confinement

Diao¹,

Ernst²,

Rawdon³

et al. 2018

J. Phys. A: Math. Theor.

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Knots in nature are typically confined spatially. The confinement affects the possible configurations, which in turn affects the spectrum of possible knot types as well as the geometry of the configurations within each knot type. The goal of this paper is to determine how confinement, length, and knotting affect the total curvature and total torsion of random polygons. Previously published papers have investigated these effects in the unconstrained case. In particular, we analyze how the total curvature and total torsion are affected by (1) varying the length of polygons within a fixed confinement radius and (2) varying the confinement radius of polygons with a fixed length. We also compare the total curvature and total torsion of groups of knots with similar complexity (measured as crossing number). While some of our results fall in line with what has been observed in the studies of the unconfined random polygons, a few surprising results emerge from our study, showing some properties that are unique due to the effect of knotting in confinement.

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“…When generating a random walk or polygon confined in a tight volume without considering the stiffness factor, the volume constrain forces the random walks and polygons to bend more. This is rather intuitive and is indeed observed in [8]. Thus the random walks and polygons generated using methods from [6,7,9] are not good candidates to model polymer chains or DNA chains that have natural stiffness since they behave very differently in terms of their geometric and topological properties than random walks and polygons with added stiffness.…”

Section: Introductionmentioning

confidence: 99%

Generating random walks and polygons with stiffness in confinement

Diao¹,

Ernst²,

Saarinen³

et al. 2015

J. Phys. A: Math. Theor.

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The purpose of this paper is to explore ways to generate random walks and polygons in confinement with a bias toward stiffness. Here the stiffness refers to the curvature angle between two consecutive edges along the random walk or polygon. The stiffer the walk (polygon), the smaller this angle on average. Thus random walks and polygons with an elevated stiffness have lower than expected curvatures. The authors introduced and studied several generation algorithms with a stiffness parameter s > 0 that regulates the expected curvature angle at a given vertex in which the random walks and polygons are generated one edge at a time using conditional probability density functions. Our generating algorithms also allow the generation of unconfined random walks and polygons with any desired mean curvature angle. In the case of random walks and polygons confined in a sphere of fixed radius, we observe that, as expected, stiff random walks or polygons are more likely to be close to the confinement boundary. The methods developed here require that the random walks and random polygons be rooted at the center of the confinement sphere.

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Curvature of random walks and random polygons in confinement

Cited by 5 publications

References 19 publications

A statistical approach to knot confinement via persistent homology

A statistical approach to knot confinement via persistent homology

Total curvature and total torsion of knotted random polygons in confinement

Generating random walks and polygons with stiffness in confinement

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