Length of the tightest trefoil knot

Baranska, Justyna; Pierański, P.; Przybyl, Sylwester; Rawdon, Eric J.

doi:10.1103/physreve.70.051810

Cited by 20 publications

(31 citation statements)

References 5 publications

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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%

“…The tightest configuration is thus given by an overlapfree, corrugated rope, centred on an equilateral polygonal knot. A preliminary version of SONO (Przybyl 2001) was followed by an improved version (Baranska et al 2004), where a finer repositioning of the vertices is implemented through an appropriate displacement towards the local centre of curvature.…”

Section: (Ii) Discretizationmentioning

confidence: 99%

“…as work on trefoil knots demonstrates (Baranska et al 2004), computational progress in numerical analysis (discretization techniques, error control, etc.) improves accuracy and this, in turn, is reflected on corrections to energy levels.…”

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confidence: 99%

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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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Section: (A) the Sono Algorithmmentioning

confidence: 99%

Section: (Ii) Discretizationmentioning

confidence: 99%

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On the groundstate energy of tight knots

Maggioni

Ricca

2009

Proc. R. Soc. A.

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“…Construction details of the polygonal knots K poly are described in previous publications [3]. Using the Rawdon method it is possible to inscribe a curvilinear knot into a polygonal knot [5].…”

Section: Discrete Representation Of Knots Tied On the Perfect Ropementioning

confidence: 99%

“…As Saka and Takano have found, the trefoil knot tied on the elastic rope subject to thermal fluctuations becomes asymmetrical: the threefold symmetry characteristic of its tight conformation [4,5] vanishes -one of its three foils becomes larger than the two other foils. One can arrive at this conclusion through an analysis of the results of studies which localise the knotted region within knots tied on long polymer filaments.…”

Section: Introductionmentioning

confidence: 99%

The average shape of the closed trefoil knot fluctuating on a floppy rope

2013

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Abstract. The average shape of the trefoil knot tied on a floppy, hard rope subject to thermal fluctuations has been determined. The fluctuations of the shape of knots were performed by random bending. As a result of the changing shape procedure large sets of deformed conformations of the initial knot were obtained. Afterwards, these sets were subject to the shape-fitting procedure. It has been found that the conformation is different from the ideal conformation of the knot.

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Scaling Behavior and Equilibrium Lengths of Knotted Polymers

et al. 2008

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We use numerical simulations to investigate how the chain length and topology of freely fluctuating knotted polymer rings affect their various spatial characteristics such as the radius of the smallest sphere enclosing momentary configurations of simulated polymer chains. We describe how the average value of a characteristic changes with the chain size and how this change depends on the topology of the modeled polymers. Although the scaling profiles of a spatial characteristic for distinct knot types do not intersect (at least, in the range of our data), the profiles for nontrivial knots intersect the corresponding profile obtained for phantom polymers, i.e., those that are free to explore all available topological states. For each knot type, this point of intersection defines its equilibrium length with respect to the spatial characteristic. At this chain length, a polymer forming a given knot type will not tend to increase or decrease, on average, the value of the spatial characteristic when the polymer is released from its topological constraint. We show interrelations between equilibrium lengths defined with respect to spatial characteristics of different character and observe that they are related to the lengths of ideal geometric configurations of the corresponding knot types.

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Length of the tightest trefoil knot

Abstract: The physical sense of tight knots provided by the SONO algorithm is discussed. A method allowing one to predict their length is presented. An upper bound for the minimum length of a smooth trefoil knot is determined.

Cited by 20 publications

References 5 publications

On the groundstate energy of tight knots

On the groundstate energy of tight knots

The average shape of the closed trefoil knot fluctuating on a floppy rope

Scaling Behavior and Equilibrium Lengths of Knotted Polymers

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