Tight open knots

Pierański, P.; Przybyl, Sylwester; Stasiak, Andrzej

doi:10.1007/s101890170012

Cited by 62 publications

(82 citation statements)

References 6 publications

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“…The extension is similar to that developed for tight open knots [67][68][69], however, it goes further to support three-periodicity. As an initial test of the algorithm, it replicates tight knot embeddings, comparable to the SONO algorithm, and we present tight embeddings of entangled θ-, tetrahedron-and cube-graphs to demonstrate its efficacy at dealing with vertices in the structure.…”

Section: Introductionsupporting

confidence: 74%

Ideal geometry of periodic entanglements

2015

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Three-dimensional entanglements, including knots, knotted graphs, periodic arrays of woven filaments and interpenetrating nets, form an integral part of structure analysis because they influence various physical properties. Ideal embeddings of these entanglements give insight into identification and classification of the geometry and physically relevant configurations in vivo. This paper introduces an algorithm for the tightening of finite, periodic and branched entanglements to a least energy form. Our algorithm draws inspiration from the ShrinkOn-No-Overlaps (SONO) (Pieranski 1998 In Ideal knots (eds A Stasiak, V Katritch, LH Kauffman), vol. 19, pp. 20-41.) algorithm for the tightening of knots and links: we call it Periodic-Branched Shrink-On-No-Overlaps (PB-SONO). We reproduce published results for ideal configurations of knots using PB-SONO. We then examine ideal geometry for finite entangled graphs, including θ-graphs and entangled tetrahedron-and cube-graphs. Finally, we compute ideal conformations of periodic weavings and entangled nets. The resulting ideal geometry is intriguing: we see spontaneous symmetrisation in some cases, breaking of symmetry in others, as well as configurations reminiscent of biological and chemical structures in nature.

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Section: Introductionsupporting

confidence: 74%

Ideal geometry of periodic entanglements

2015

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“…Similar to the previous study, 28 we define p as the ratio of (2) where L ideal is a topological property defined for a maximally inflated knot. 32 We then obtain…”

Section: Introductionmentioning

confidence: 99%

Metastable Tight Knots in Semiflexible Chains

2014

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Knotted structures can spontaneously occur in polymers such as DNA and proteins, and the formation of knots affects biological functions, mechanical strength and rheological properties. In this work, we calculate the equilibrium size distribution of trefoil knots in linear DNA using off-lattice simulations. We observe metastable knots on DNA, as predicted by Grosberg and Rabin. Furthermore, we extend their theory to incorporate the finite width of chains and show an agreement between our simulations and the modified theory for real chains. Our results suggest localized knots spontaneously occur in long DNA and the contour length in the knot ranges from 600 to 1800 nm.

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“…Technical details of this algorithm are widely available in the literature; the interested reader can consult several papers such as Pieranski et al (2001), Pieranski & Przybyl (2002) and Baranska et al (2004Baranska et al ( , 2005. We present here the ideas behind the algorithm and we highlight some critical aspects.…”

Section: (A) the Sono Algorithmmentioning

confidence: 99%

On the groundstate energy of tight knots

Maggioni

Ricca

2009

Proc. R. Soc. A.

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New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume-and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots -up to 10 crossings -is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.

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Tight open knots

Abstract: The most tight conformations of prime knots are found with the use of the SONO algorithm. Their curvature and torsion profiles are calculated. Symmetry of the knots is analysed. Connections with the physics of polymers are discussed.

Cited by 62 publications

References 6 publications

Ideal geometry of periodic entanglements

Ideal geometry of periodic entanglements

Metastable Tight Knots in Semiflexible Chains

On the groundstate energy of tight knots

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