Search results for "Geometric topology"

2016

We determine knotting probabilities and typical sizes of knots in double-stranded DNA for chains of up to half a million base pairs with computer simulations of a coarse-grained bead-stick model: Single trefoil knots and composite knots which include at least one trefoil as a prime factor are shown to be common in DNA chains exceeding 250,000 base pairs, assuming physiologically relevant salt conditions. The analysis is motivated by the emergence of DNA nanopore sequencing technology, as knots are a potential cause of erroneous nucleotide reads in nanopore sequencing devices and may severely limit read lengths in the foreseeable future. Even though our coarse-grained model is only based on …

KnotGenome: a server to analyze entanglements of chromosomes.

2018

Abstract The KnotGenome server enables the topological analysis of chromosome model data using three-dimensional coordinate files of chromosomes as input. In particular, it detects prime and composite knots in single chromosomes, and links between chromosomes. The knotting complexity of the chromosome is presented in the form of a matrix diagram that reveals the knot type of the entire polynucleotide chain and of each of its subchains. Links are determined by means of the Gaussian linking integral and the HOMFLY-PT polynomial. Entangled chromosomes are presented graphically in an intuitive way. It is also possible to relax structure with short molecular dynamics runs before the analysis. Kn…

The proof of Birman’s conjecture on singular braid monoids

2003

Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birman's conjecture, namely, that the desingularization map eta is injective.

Spectral sequence associated with a symplectic manifold

2006

A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C-spectral sequence is indicated.

Simple connections between generalized hypergeometric series and dilogarithms

1997

AbstractConnections between generalized hypergeometric series and dilogarithms are investigated. Some simple relations of an Appell's function and dilogarithms are found.

On hyperbolic type involutions

2001

We give a bound on the number of hyperbolic knots which are double covered by a fixed (non hyperbolic) manifold in terms of the number of tori and of the invariants of the Seifert fibred pieces of its Jaco-Shalen-Johannson decomposition. We also investigate the problem of finding the non hyperbolic knots with the same double cover of a hyperbolic one and give several examples to illustrate the results.

On deformation of Poisson manifolds of hydrodynamic type

2001

We study a class of deformations of infinite-dimensional Poisson manifolds of hydrodynamic type which are of interest in the theory of Frobenius manifolds. We prove two results. First, we show that the second cohomology group of these manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial. Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous formal deformations of the above manifolds.

Roots in the mapping class groups

2006

The purpose of this paper is the study of the roots in the mapping class groups. Let $\Sigma$ be a compact oriented surface, possibly with boundary, let $\PP$ be a finite set of punctures in the interior of $\Sigma$, and let $\MM (\Sigma, \PP)$ denote the mapping class group of $(\Sigma, \PP)$. We prove that, if $\Sigma$ is of genus 0, then each $f \in \MM (\Sigma)$ has at most one $m$-root for all $m \ge 1$. We prove that, if $\Sigma$ is of genus 1 and has non-empty boundary, then each $f \in \MM (\Sigma)$ has at most one $m$-root up to conjugation for all $m \ge 1$. We prove that, however, if $\Sigma$ is of genus $\ge 2$, then there exist $f,g \in \MM (\Sigma, \PP)$ such that $f^2=g^2$, $…

Knot Theory, Jones Polynomial and Quantum Computing

2005

Knot theory emerged in the nineteenth century for needs of physics and chemistry as these needs were understood those days. After that the interest of physicists and chemists was lost for about a century. Nowadays knot theory has made a comeback. Knot theory and other areas of topology are no more considered as abstract areas of classical mathematics remote from anything of practical interest. They have made deep impact on quantum field theory, quantum computation and complexity of computation.

KNOTS WITH UNKNOTTING NUMBER ONE AND GENERALISED CASSON INVARIANT

1996

We extend the classical notion of unknotting operation to include operations on rational tangles. We recall the “classical” conditions (on the signature, linking form etc.) for a knot to have integral (respectively rational) unknotting number one. We show that the generalised Casson invariant of the twofold branched cover of the knot gives a further necessary condition. We apply these results to some Montesinos knots and to knots with less than nine crossings.