Characterization of entanglements in glassy polymeric ensembles using the Gaussian linking number

Ahmad, Rasool; Paul, Saptarshi; Basu, S.

doi:10.1103/physreve.101.022503

Cited by 13 publications

(9 citation statements)

References 40 publications

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“…Such use of the Gaussian linking integral as an order parameter for the entanglement of open curves was first explored in reference [42] for uniform random walks and equilateral random polygons confined in space (see sections 4.2 and 4.3). The same measure was also used for pairwise entanglement in polymer melts and solutions [43][44][45] and in biological-motivated contexts, such as detecting interlocked chains in multimeric proteins [46,47].…”

Section: Physical Linksmentioning

confidence: 99%

Topological and physical links in soft matter systems

Orlandini

Micheletti

2021

J. Phys.: Condens. Matter

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Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packed in vivo to interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.

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Section: Physical Linksmentioning

confidence: 99%

Topological and physical links in soft matter systems

Orlandini

Micheletti

2021

J. Phys.: Condens. Matter

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“…ere are many applications of the linking number in different disciplines [46,47]. Correspondingly, the self-linking number for cycle A, which represents the linking of a periodic orbit with itself, can be calculated as Complexity…”

Section: Symbolic Encoding Of Periodic Orbits With Four Letters Formentioning

confidence: 99%

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Dong

Jia

et al. 2021

Complexity

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To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

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“…This integer degree is the linking number of the 2-component link 𝛾 1 𝛾 2 ⊂ R 3 formed by the two closed curves. Invariance modulo continuous deformation of R 3 follows easily for closed curves -indeed, the function under the Gauss integral in (1), and hence the integral itself, varies continuously under perturbations of the curves 𝛾 1 , 𝛾 2 . This should keep any integer value constant.…”

Section: The Gauss Integral For the Linking Number Of Curvesmentioning

confidence: 99%

“…In R 3 with the Euclidean metric these are rotations, translations and reflections. The isometry invariance of the real-valued linking number for open curves has found applications in the study of molecules [1].…”

Section: The Gauss Integral For the Linking Number Of Curvesmentioning

confidence: 99%

A Proof of the Invariant Based Formula for the Linking Number and its Asymptotic Behaviour

Bright¹,

Anosova²,

Kurlin³

2020

Preprint

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In 1833 Gauss defined the linking number of two disjoint curves in 3-space. For open curves this double integral over the parameterised curves is real-valued and invariant modulo rigid motions or isometries that preserve distances between points, and has been recently used in the elucidation of molecular structures. In 1976 Banchoff geometrically interpreted the linking number between two line segments. An explicit analytic formula based on this interpretation was given in 2000 without proof in terms of 6 isometry invariants: the distance and angle between the segments and 4 coordinates specifying their relative positions. We give a detailed proof of this formula and describe its asymptotic behaviour that wasn't previously studied.

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Characterization of entanglements in glassy polymeric ensembles using the Gaussian linking number

Cited by 13 publications

References 40 publications

Topological and physical links in soft matter systems

Topological and physical links in soft matter systems

Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

A Proof of the Invariant Based Formula for the Linking Number and its Asymptotic Behaviour

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