82 research outputs found
Ropelength of tight polygonal knots
A physical interpretation of the rope simulated by the SONO algorithm is
presented. Properties of the tight polygonal knots delivered by the algorithm
are analyzed. An algorithm for bounding the ropelength of a smooth inscribed
knot is shown. Two ways of calculating the ropelength of tight polygonal knots
are compared. An analytical calculation performed for a model knot shows that
an appropriately weighted average should provide a good estimation of the
minimum ropelength for relatively small numbers of edges.Comment: 27 pages, to appear in "Physical and Numerical Models in Knot Theory
and their Application to the Life Sciences
Knot Tightening By Constrained Gradient Descent
We present new computations of approximately length-minimizing polygons with
fixed thickness. These curves model the centerlines of "tight" knotted tubes
with minimal length and fixed circular cross-section. Our curves approximately
minimize the ropelength (or quotient of length and thickness) for polygons in
their knot types. While previous authors have minimized ropelength for polygons
using simulated annealing, the new idea in our code is to minimize length over
the set of polygons of thickness at least one using a version of constrained
gradient descent.
We rewrite the problem in terms of minimizing the length of the polygon
subject to an infinite family of differentiable constraint functions. We prove
that the polyhedral cone of variations of a polygon of thickness one which do
not decrease thickness to first order is finitely generated, and give an
explicit set of generators. Using this cone we give a first-order minimization
procedure and a Karush-Kuhn-Tucker criterion for polygonal ropelength
criticality.
Our main numerical contribution is a set of 379 almost-critical prime knots
and links, covering all prime knots with no more than 10 crossings and all
prime links with no more than 9 crossings. For links, these are the first
published ropelength figures, and for knots they improve on existing figures.
We give new maps of the self-contacts of these knots and links, and discover
some highly symmetric tight knots with particularly simple looking self-contact
maps.Comment: 45 pages, 16 figures, includes table of data with upper bounds on
ropelength for all prime knots with no more than 10 crossings and all prime
links with no more than 9 crossing
Knot Fertility and Lineage
In this paper, we introduce a new type of relation between knots called the
descendant relation. One knot is a descendant of another knot if
can be obtained from a minimal crossing diagram of by some number of
crossing changes. We explore properties of the descendant relation and study
how certain knots are related, paying particular attention to those knots,
called fertile knots, that have a large number of descendants. Furthermore, we
provide computational data related to various notions of knot fertility and
propose several open questions for future exploration.Comment: 20 pages, 11 figures, 14 table
Tight Knot Spectrum In Qcd
We model the observed J(++) mesonic mass spectrum in terms of energies for tightly knotted and linked chromoelectric QCD flux tubes. The data is fit with one- and two-parameter models. We predict a possible new state at approximately 1190 MeV and a plethora of new states above 1690 MeV
Shapes of Knotted Cyclic Polymers(Knots and soft-matter physics: Topology of polymers and related topics in physics, mathematics and biology)
この論文は国立情報学研究所の電子図書館事業により電子化されました。Momentary configurations of long polymers at thermal equilibrium usually deviate from spherical symmetry and can be better described, on average, by a prolate ellipsoid. The asphericity and nature of asphericity (or prolateness) that describe these momentary ellipsoidal shapes of a polymer are determined by specific expressions involving the three principal moments of inertia calculated for configurations of the polymer. Earlier theoretical studies and numerical simulations have established that as the length of the polymer increases, the average shape for the statistical ensemble of random configurations asymptotically approaches a characteristic universal shape that depends on the solvent quality. It has been established, however, that these universal shapes differ for linear, circular, and branched chains. We investigate here the effect of knotting on the shape of cyclic polymers modeled as random isosegmental polygons. We observe that random polygons forming different knot types reach asymptotic shapes that are distinct from the ensemble average shape. For the same chain length, more complex knots are, on average, more spherical than less complex knots. This paper is a shorter, revised version of the article Ref. [12]. For more details, see Ref. [12]
Bending modes of DNA directly addressed by cryo-electron microscopy of DNA minicircles
We use cryo-electron microscopy (cryo-EM) to study the 3D shapes of 94-bp-long DNA minicircles and address the question of whether cyclization of such short DNA molecules necessitates the formation of sharp, localized kinks in DNA or whether the necessary bending can be redistributed and accomplished within the limits of the elastic, standard model of DNA flexibility. By comparing the shapes of covalently closed, nicked and gapped DNA minicircles, we conclude that 94-bp-long covalently closed and nicked DNA minicircles do not show sharp kinks while gapped DNA molecules, containing very flexible single-stranded regions, do show sharp kinks. We corroborate the results of cryo-EM studies by using Bal31 nuclease to probe for the existence of kinks in 94-bp-long minicircle
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