128 research outputs found
Knots, Braids and Hedgehogs from the Eikonal Equation
The complex eikonal equation in the three space dimensions is considered. We
show that apart from the recently found torus knots this equation can also
generate other topological configurations with a non-trivial value of the
index: braided open strings as well as hedgehogs. In particular,
cylindric strings i.e. string solutions located on a cylinder with a constant
radius are found. Moreover, solutions describing strings lying on an arbitrary
surface topologically equivalent to cylinder are presented. We discus them in
the context of the eikonal knots. The physical importance of the results
originates in the fact that the eikonal knots have been recently used to
approximate the Faddeev-Niemi hopfions.Comment: 15 pages, 5 figure
Topological effects in the thermal properties of knotted polymer rings
The topological effects on the thermal properties of several knot
configurations are investigated using Monte Carlo simulations. In order to
check if the topology of the knots is preserved during the thermal fluctuations
we propose a method that allows very fast calculations and can be easily
applied to arbitrarily complex knots. As an application, the specific energy
and heat capacity of the trefoil, the figure-eight and the knots are
calculated at different temperatures and for different lengths. Short-range
repulsive interactions between the monomers are assumed. The knots
configurations are generated on a three-dimensional cubic lattice and sampled
by means of the Wang-Landau algorithm and of the pivot method. The obtained
results show that the topological effects play a key role for short-length
polymers. Three temperature regimes of the growth rate of the internal energy
of the system are distinguished.Comment: 7 pages, 12 figures, LaTeX + RevTeX. With respect to the first
version, in the second version the text has been improved and all figures are
now in black and whit
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Equilibrium size of large ring molecules
The equilibrium properties of isolated ring molecules were investigated using
an off-lattice model with no excluded volume but with dynamics that preserve
the topological class. Using an efficient set of long range moves, chains of
more than 2000 monomers were studied. Despite the lack of any excluded volume
interaction, the radius of gyration scaled like that of a self avoiding walk,
as had been previously conjectured. However this scaling was only seen for
chains greater than 500 monomers.Comment: 11 pages, 3 eps figures, latex, psfi
Knot localization in adsorbing polymer rings
We study by Monte Carlo simulations a model of knotted polymer ring adsorbing
onto an impenetrable, attractive wall. The polymer is described by a
self-avoiding polygon (SAP) on the cubic lattice. We find that the adsorption
transition temperature, the crossover exponent and the metric exponent
, are the same as in the model where the topology of the ring is
unrestricted. By measuring the average length of the knotted portion of the
ring we are able to show that adsorbed knots are localized. This knot
localization transition is triggered by the adsorption transition but is
accompanied by a less sharp variation of the exponent related to the degree of
localization. Indeed, for a whole interval below the adsorption transition, one
can not exclude a contiuous variation with temperature of this exponent. Deep
into the adsorbed phase we are able to verify that knot localization is strong
and well described in terms of the flat knot model.Comment: 27 pages, 10 figures. Submitter to Phys. Rev.
Minimal knotted polygons in cubic lattices
An implementation of BFACF-style algorithms on knotted polygons in the simple
cubic, face centered cubic and body centered cubic lattice is used to estimate
the statistics and writhe of minimal length knotted polygons in each of the
lattices. Data are collected and analysed on minimal length knotted polygons,
their entropy, and their lattice curvature and writhe
The Compressibility of Minimal Lattice Knots
The (isothermic) compressibility of lattice knots can be examined as a model
of the effects of topology and geometry on the compressibility of ring
polymers. In this paper, the compressibility of minimal length lattice knots in
the simple cubic, face centered cubic and body centered cubic lattices are
determined. Our results show that the compressibility is generally not
monotonic, but in some cases increases with pressure. Differences of the
compressibility for different knot types show that topology is a factor
determining the compressibility of a lattice knot, and differences between the
three lattices show that compressibility is also a function of geometry.Comment: Submitted to J. Stat. Mec
Abundance of unknots in various models of polymer loops
A veritable zoo of different knots is seen in the ensemble of looped polymer
chains, whether created computationally or observed in vitro. At short loop
lengths, the spectrum of knots is dominated by the trivial knot (unknot). The
fractional abundance of this topological state in the ensemble of all
conformations of the loop of segments follows a decaying exponential form,
, where marks the crossover from a mostly unknotted
(ie topologically simple) to a mostly knotted (ie topologically complex)
ensemble. In the present work we use computational simulation to look closer
into the variation of for a variety of polymer models. Among models
examined, is smallest (about 240) for the model with all segments of the
same length, it is somewhat larger (305) for Gaussian distributed segments, and
can be very large (up to many thousands) when the segment length distribution
has a fat power law tail.Comment: 13 pages, 6 color figure
On the Dominance of Trivial Knots among SAPs on a Cubic Lattice
The knotting probability is defined by the probability with which an -step
self-avoiding polygon (SAP) with a fixed type of knot appears in the
configuration space. We evaluate these probabilities for some knot types on a
simple cubic lattice. For the trivial knot, we find that the knotting
probability decays much slower for the SAP on the cubic lattice than for
continuum models of the SAP as a function of . In particular the
characteristic length of the trivial knot that corresponds to a `half-life' of
the knotting probability is estimated to be on the cubic
lattice.Comment: LaTeX2e, 21 pages, 8 figur
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