68 research outputs found
Geometry and topology of knotted ring polymers in an array of obstacles
We study knotted polymers in equilibrium with an array of obstacles which
models confinement in a gel or immersion in a melt. We find a crossover in both
the geometrical and the topological behavior of the polymer. When the polymers'
radius of gyration, , and that of the region containing the knot,
, are small compared to the distance b between the obstacles, the knot
is weakly localised and scales as in a good solvent with an amplitude
that depends on knot type. In an intermediate regime where ,
the geometry of the polymer becomes branched. When exceeds b, the
knot delocalises and becomes also branched. In this regime, is
independent of knot type. We discuss the implications of this behavior for gel
electrophoresis experiments on knotted DNA in weak fields.Comment: 4 pages, 6 figure
On the size of knots in ring polymers
We give two different, statistically consistent definitions of the length l
of a prime knot tied into a polymer ring. In the good solvent regime the
polymer is modelled by a self avoiding polygon of N steps on cubic lattice and
l is the number of steps over which the knot ``spreads'' in a given
configuration. An analysis of extensive Monte Carlo data in equilibrium shows
that the probability distribution of l as a function of N obeys a scaling of
the form p(l,N) ~ l^(-c) f(l/N^D), with c ~ 1.25 and D ~ 1. Both D and c could
be independent of knot type. As a consequence, the knot is weakly localized,
i.e. ~ N^t, with t=2-c ~ 0.75. For a ring with fixed knot type, weak
localization implies the existence of a peculiar characteristic length l^(nu) ~
N^(t nu). In the scaling ~ N^(nu) (nu ~0.58) of the radius of gyration of the
whole ring, this length determines a leading power law correction which is much
stronger than that found in the case of unrestricted topology. The existence of
such correction is confirmed by an analysis of extensive Monte Carlo data for
the radius of gyration. The collapsed regime is studied by introducing in the
model sufficiently strong attractive interactions for nearest neighbor sites
visited by the self-avoiding polygon. In this regime knot length determinations
can be based on the entropic competition between two knotted loops separated by
a slip link. These measurements enable us to conclude that each knot is
delocalized (t ~ 1).Comment: 29 pages, 14 figure
Knot invariants in lens spaces
In this survey we summarize results regarding the Kauffman bracket, HOMFLYPT,
Kauffman 2-variable and Dubrovnik skein modules, and the Alexander polynomial
of links in lens spaces, which we represent as mixed link diagrams. These
invariants generalize the corresponding knot polynomials in the classical case.
We compare the invariants by means of the ability to distinguish between some
difficult cases of knots with certain symmetries
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.
An Inquiry into the Practice of Proving in Low-Dimensional Topology
The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used in the practice are an integral part of the mathematical reasoning. As a matter of fact, they convey in a material form the relevant transitions and thus allow experts to draw inferential connections. Second, in low-dimensional topology experts exploit a particular type of manipulative imagination which is connected to intuition of two- and three-dimensional space and motor agency. This imagination allows recognizing the transformations which connect different pictures in an argument. Third, the epistemic—and inferential—actions performed are permissible only within a specific practice: this form of reasoning is subject-matter dependent. Local criteria of validity are established to assure the soundness of representationally heterogeneous arguments in low-dimensional topology
Knotlike Cosmic Strings in The Early Universe
In this paper, the knotlike cosmic strings in the Riemann-Cartan space-time
of the early universe are discussed. It has been revealed that the cosmic
strings can just originate from the zero points of the complex scalar
quintessence field. In these strings we mainly study the knotlike
configurations. Based on the integral of Chern-Simons 3-form a topological
invariant for knotlike cosmic strings is constructed, and it is shown that this
invariant is just the total sum of all the self-linking and linking numbers of
the knots family. Furthermore, it is also pointed out that this invariant is
preserved in the branch processes during the evolution of cosmic strings
Quantum Knitting
We analyze the connections between the mathematical theory of knots and
quantum physics by addressing a number of algorithmic questions related to both
knots and braid groups.
Knots can be distinguished by means of `knot invariants', among which the
Jones polynomial plays a prominent role, since it can be associated with
observables in topological quantum field theory.
Although the problem of computing the Jones polynomial is intractable in the
framework of classical complexity theory, it has been recently recognized that
a quantum computer is capable of approximating it in an efficient way. The
quantum algorithms discussed here represent a breakthrough for quantum
computation, since approximating the Jones polynomial is actually a `universal
problem', namely the hardest problem that a quantum computer can efficiently
handle.Comment: 29 pages, 5 figures; to appear in Laser Journa
The Combinatorics of Alternating Tangles: from theory to computerized enumeration
We study the enumeration of alternating links and tangles, considered up to
topological (flype) equivalences. A weight is given to each connected
component, and in particular the limit yields information about
(alternating) knots. Using a finite renormalization scheme for an associated
matrix model, we first reduce the task to that of enumerating planar
tetravalent diagrams with two types of vertices (self-intersections and
tangencies), where now the subtle issue of topological equivalences has been
eliminated. The number of such diagrams with vertices scales as for
. We next show how to efficiently enumerate these diagrams (in time
) by using a transfer matrix method. We give results for various
generating functions up to 22 crossings. We then comment on their large-order
asymptotic behavior.Comment: proceedings European Summer School St-Petersburg 200
"New" Veneziano amplitudes from "old" Fermat (hyper) surfaces
The history of discovery of bosonic string theory is well documented. This
theory evolved as an attempt to find a multidimensional analogue of Euler's
beta function. Such an analogue had in fact been known in mathematics
literature at least in 1922 and was studied subsequently by mathematicians such
as Selberg, Weil and Deligne among others. The mathematical interpretation of
this multidimensional beta function is markedly different from that described
in physics literature. This paper aims to bridge the gap between the existing
treatments. Preserving all results of conformal field theories intact,
developed formalism employing topological, algebro-geometric, number-theoretic
and combinatorial metods is aimed to provide better understanding of the
Veneziano amplitudes and, thus, of string theories.Comment: 92 pages LaTex, some typos removed, discussion section is added along
with several additional latest reference
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