The Generation of Random Equilateral Polygons

Alvarado, Sotero; Calvo, Jorge Alberto; Millett, Kenneth C.

doi:10.1007/s10955-011-0164-4

Cited by 23 publications

(44 citation statements)

References 65 publications

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“…There are a number of methods available for the random generation of equilateral knots [1]. Because we seek to have the arc close to give a ring polygon, we can't merely generate an arbitrary random walk.…”

Section: Previous Thickness Free Generation Methodsmentioning

confidence: 99%

“…Most methods come in two stages, an initialization stage, and a randomization stage. Two similar methods for initializing a closed polygon are the generalized hedgehog method, and the triangle method [1]. These involve randomly generating sets of two or three vectors which add to zero, and then randomly ordering these vectors.…”

Section: Previous Thickness Free Generation Methodsmentioning

confidence: 99%

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An ergodic algorithm for generating knots with a prescribed injectivity radius

Chapman

2019

J. Knot Theory Ramifications

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The first algorithm for sampling the space of thick equilateral knots, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It also is an off lattice equivalent of the pivot algorithm. To prove the efficacy of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint and so is ergodic. This ergodic sampling unlocks the ability to analyze the effects of thickness on properties of the geometric knot such as radius of gyration and probability of unknotting.

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Section: Previous Thickness Free Generation Methodsmentioning

confidence: 99%

Section: Previous Thickness Free Generation Methodsmentioning

confidence: 99%

An ergodic algorithm for generating knots with a prescribed injectivity radius

Chapman

2019

J. Knot Theory Ramifications

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“…In order to simulate an Olympic gel, we first create an unknotted random 125 edge equilateral polygon using the crankshaft algorithm [1]. The resulting polygon is randomly placed in the interior of a generating cell and extended to a system with 1, 2, or 3 PBC.…”

Section: Simulation Of Olympic Gel Systemsmentioning

confidence: 99%

Resolving critical degrees of entanglement in Olympic ring systems

Igram

Millett

Panagiotou

2016

J. Knot Theory Ramifications

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Olympic systems are collections of small ring polymers whose aggregate properties are largely characterized by the extent (or absence) of topological linking in contrast with the topological entanglement arising from physical movement constraints associated with excluded volume contacts or arising from chemical bonds. First, discussed by de Gennes, they have been of interest ever since due to their particular properties and their occurrence in natural organisms, for example, as intermediates in the replication of circular DNA in the mitochondria of malignant cells or in the kinetoplast DNA networks of trypanosomes. Here, we study systems that have an intrinsic one, two, or threedimensional character and consist of large collections of ring polymers modeled using periodic boundary conditions. We identify and discuss the evolution of the dimensional character of the large scale topological linking as a function of density. We identify the critical densities at which infinite linked subsystems, the onset of percolation, arise in the periodic boundary condition systems. These provide insight into the nature of entanglement occurring in such course grained models. This entanglement is measured using Gauss linking number, a measure well adapted to such models. We show that, with increasing density, the topological entanglement of these systems increases in complexity, dimension, and probability.

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“…al. 36 have shown that a conservative sampling rate of 4n crankshaft rotationsis adequate for polygons through 120 edges. In fact, a sampling rate of n crankshaft rotations is sufficient to achieve PCI values within one standard deviation of the random distribution without the closure condition.…”

Section: Measures Of Randomnessmentioning

confidence: 99%

Physical Knot Theory: An Introduction to the Study of the Influence of Knotting on the Spatial Characteristics of Polymers

Millett

2011

Introductory Lectures on Knot Theory

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This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.

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The Generation of Random Equilateral Polygons

Cited by 23 publications

References 65 publications

An ergodic algorithm for generating knots with a prescribed injectivity radius

An ergodic algorithm for generating knots with a prescribed injectivity radius

Resolving critical degrees of entanglement in Olympic ring systems

Physical Knot Theory: An Introduction to the Study of the Influence of Knotting on the Spatial Characteristics of Polymers

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