Sequence Of Knots Research Articles

Twist moves and the affine index polynomials of virtual knots

In this paper, we give a necessary condition for two virtual knots to be related by a finite sequence of twist moves by using the affine index polynomial, which is a Vassiliev invariant of degree [Formula: see text]. Trapp showed that a numerical Vassiliev invariant of degree [Formula: see text] has a polynomial growth of degree [Formula: see text] on a twist sequence of knots, which can be extended to a twist sequence of virtual knots. We calculate the growth of the affine index polynomial for a twist sequence of virtual knots and find the difference of the affine index polynomials of two virtual knots, which are related by a twist move. Moreover, we give a lower bound for the number of twist moves needed to transform [Formula: see text] to [Formula: see text] if [Formula: see text] and [Formula: see text] are virtual knots related by a finite sequence of twist moves.

On the optimization of knot allocation for B-spline parameterization of the dielectric function in spectroscopic ellipsometry data analysis

Dielectric function representation by B-splines became quite popular and widely used in the context of spectroscopic ellipsometry data interpretation. B-splines are defined by a polynomial degree and a sequence of knots (i.e., the number and positions of knots). Defining the knot sequence is non-trivial, and this task has a significant effect on the actual effectiveness of the B-spline parameterization in spectroscopic ellipsometry data analysis. In this paper, we propose a simple, practical, and systematic knot placement scheme that improves ordinary trial-and-error technique in establishing the knot spacing. The approach suggested here is based on an integral span, a measure introduced in this work. The proposed procedure provides a possibility to determine the knot locations automatically (or, at least, semi-automatically) and excludes widespread modeling ambiguities associated with uncertain knot vector. Moreover, our approach guarantees the absence of non-monotonic behavior of the mean-squared-error function and thereby improves the accuracy of our analysis. The performance of the proposed method has been tested for several real cases of the study.

Chebotarev links are stably generic

We discuss the relationship between two analogues in a 3-manifold of the set of prime ideals in a number field. We prove that if $(K_i)_{i\in \mathbb{N}_{>0}}$ is a sequence of knots obeying the Chebotarev law in the sense of Mazur and McMullen, then $\mathcal{K}=\cup_i K_i$ is a stably generic link in the sense of Mihara. An example we investigate is the planetary link of a fibered hyperbolic finite link in $S^3$. We also observe a Chebotarev phenomenon of knot decomposition in a degree 5 non-Galois subcover of an $A_5$(icosahedral)-cover.

Spacious knots

We show that there exist hyperbolic knots in the 3-sphere such that the set of points of large injectivity radius in the complement take up the bulk of the volume. More precisely, given a finite volume hyperbolic manifold, for any bound R>0 on injectivity radius, consider the set of points with injectivity radius at least R; we call this the R-thick part of the manifold. We show that for any $\epsilon>0$, there exists a knot K in the 3-sphere so that the ratio of the volume of the R-thick part of the knot complement to the volume of the knot complement is at least $1-\epsilon$. As R approaches infinity, and as $\epsilon$ approaches zero, this gives a sequence of knots that is said to Benjamini--Schramm converge to hyperbolic space. This answers a question of Brock and Dunfield.

HOMFLYPT polynomial is the best quantifier for topological cascades of vortex knots

In this paper we derive and compare numerical sequences obtained by adapted polynomials such as HOMFLYPT, Jones and Alexander-Conway for the topological cascade of vortex torus knots and links that progressively untie by a single reconnection event at a time. Two cases are considered: the alternate sequence of knots and co-oriented links (with positive crossings) and the sequence of two-component links with oppositely oriented components (negative crossings). New recurrence equations are derived and sequences of numerical values are computed. In all cases the adapted HOMFLYPT polynomial proves to be the best quantifier for the topological cascade of torus knots and links.

Independence of volume and genus 𝑔 bridge numbers

A theorem of Jorgensen and Thurston implies that the volume of a hyperbolic 3–manifold is bounded below by a linear function of its Heegaard genus. Heegaard surfaces and bridge surfaces often exhibit similar topological behavior; thus it is natural to extend this comparison to ask whether a ( g , b ) (g,b) -bridge surface for a knot K K in S 3 S^3 carries any geometric information related to the knot exterior. In this paper, we show that — unlike in the case of Heegaard splittings — hyperbolic volume and genus g g -bridge numbers are completely independent. That is, for any g g , we construct explicit sequences of knots with bounded volume and unbounded genus g g -bridge number, and explicit sequences of knots with bounded genus g g -bridge number and unbounded volume.

Density spectra for knots

We recently discovered a relationship between the volume density spectrum and the determinant density spectrum for infinite sequences of hyperbolic knots. Here, we extend this study to new quantum density spectra associated to quantum invariants, such as Jones polynomials, Kashaev invariants and knot homology. We also propose related conjectures motivated by geometrically and diagrammatically maximal sequences of knots.

How Bad Are Vandermonde Matrices?

Estimation of the condition numbers of Vandermonde matrices, motivated by applications to interpolation and quadrature, can be traced back to at least the 1970s. Empirical study has consistently shown that Vandermonde matrices tend to be badly ill-conditioned, with a narrow class of exceptions, such as the matrices of the discrete Fourier transform (hereafter we use the acronym DFT). So far, however, formal support for this empirical observation has been limited to the matrices defined by the real set of knots. We prove that, more generally, any Vandermonde matrix of a large size is badly ill-conditioned unless its knots are more or less equally spaced on or about the circle $\{x:~|x|=1\}$. The matrices of the DFT are unitary up to scaling and therefore perfectly conditioned. They are defined by a cyclic sequence of knots, equally spaced on that circle, but we prove that even a slight modification of the knots into the so-called quasi-cyclic sequence on this circle defines badly ill-conditioned Vandermonde matrices. Likewise, we prove that the half-size leading (that is, northwestern) block of a large DFT matrix is badly ill-conditioned. (This result was motivated by an application to preconditioning of an input matrix for Gaussian elimination with no pivoting and for block Gaussian elimination.) Our analysis involves the Ekkart--Young theorem, the Vandermonde-to-Cauchy transformation of matrix structure, our new inversion formula for a Cauchy matrix, and low-rank approximation of its large submatrices.

MOLECULAR JET OF IRAS 04166+2706

The molecular outflow from IRAS 04166+2706 was mapped with the Submillimeter Array (SMA) at 350 GHz continuum and CO J = 3$-$2 at an angular resolution of ~1 arcsec. The field of view covers the central arc-minute, which contains the inner four pairs of knots of the molecular jet. On the channel map, conical structures are clearly present in the low velocity range (|V$-$V$_0$|$<$10 km s$^{-1}$), and the highly collimated knots appear in the Extremely High Velocity range (EHV, 50$>$|V$-$V$_0$|$>$30 km s$^{-1}$). The higher angular resolution of ~1 arcsec reveals the first blue-shifted knot (B1) that was missing in previous PdBI observation of Sant\'iago-Garc\'ia et al. (2009) at an offset of ~6 arcsec to the North-East of the central source. This identification completes the symmetric sequence of knots in both the blue- and red-shifted lobes of the outflow. The innermost knots R1 and B1 have the highest velocities within the sequence. Although the general features appear to be similar to previous CO J = 2$-$1 images in Sant\'iago-Garc\'ia et al. (2009), the emission in CO J = 3$-$2 almost always peaks further away from the central source than that of CO J = 2$-$1 in the red-shifted lobe of the channel maps. This gives rise to a gradient in the line-ratio map of CO J = 3$-$2/J = 2$-$1 from head to tail within a knot. A large velocity gradient (LVG) analysis suggests that the differences may reflect a higher gas kinetic temperature at the head. We also explore possible constraints imposed by the non-detection of SiO J = 8$-$7.

Splines of Complex Order: Fourier, Filters and Fractional Derivatives

For the Schoenberg (polynomial) B-splines, interesting relations between their functional representation, Dirichlet averages, and difference operators are known. We use these relations to extend B-splines to an arbitrary (infinite) sequence of knots and to higher dimensions. The Fourier domain representation of multidimensional complex B-splines, their two-scale relation and an explicit time domain representation are given for the case of equidistantly distributed knots on rays. In addition, we show that complex B-splines are solutions of the sampling/interpolation problem $${D^z}f = \sum\limits_{k = 0}^\infty {{a_k}\delta ( \bullet - k)}$$ of complex order z, Re z > 1.

Free Knot Splines with RJMCMC in Survival Data Analysis

The B-spline representation is a common tool to improve the fitting of smooth nonlinear functions, it offers a fitting as a piecewise polynomial. The regions that define the pieces are separated by a sequence of knots. The main difficulty in this type of modeling is the choice of the number and the locations of these knots. The Reversible Jump Markov Chain Monte Carlo (RJMCMC) algorithm provides a solution to simultaneously select these two parameters by considering the knots as free parameters. This algorithm belongs to the MCMC techniques that allow simulations from target distributions on spaces of varying dimension. The aim of the present investigation is to use this algorithm in the framework of the analysis of survival time, for the Cox model in particular. In fact, the relation between the hazard ratio function and the covariates being assumed to be log-linear, this assumption is too restrictive. Thus, we propose to use the RJMCMC algorithm to model the log hazard ratio function by a B-spline representation with an unknown number of knots at unknown locations. This method is illustrated with two real data sets: the Stanford heart transplant data and lung cancer survival data. Another application of the RJMCMC is selecting the significant covariates, and a simulation study is performed.

B-Spline Model Order Selection With Optimal MDL Criterion Applied to Speech Fundamental Frequency Stylization

In the speech processing field, stylization of fundamental frequency <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> has been subjected to numerous works. Models proposed in the literature rely on knowledge stemming from phonology and linguistics. We propose an approach that deals with the issue of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> curve stylization requiring as few linguistic assumptions as possible and in the framework of B-spline models. A B-spline model, characterized by a sequence of knots with which control points are associated, enables the formalization of discontinuities in the derivatives of the observed values sequence. Beyond the implementation of a B-spline model to stylize an open curve sampled using a constant step, we address the problem of the optimal model order choice. We propose to use a parsimony criterion based on a minimum description length (MDL) approach, in order to optimize the number of knots. We derive several criteria relying on bounds estimated from parameter values. We demonstrate the optimality of these choices in the theoretical MDL framework. We introduce a notion of variable precision of parameters which enables a good compromise between the modeling precision and degrees of freedom of the estimated models. Experiments are performed on a French speech corpus and compare three MDL criteria. The use of both B-spline model and MDL methodology enables an efficient modeling of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> curves and provides an RMS error around 1 Hz while allowing a relatively high compression rate about 40%.

Interpolation in Bernstein and Paley–Wiener spaces

We consider interpolation of discrete functions by continuous ones with restriction on the size of spectra. We discuss a sharp contrast between the cases of compact and unbounded spectra. In particular we construct ‘universal’ spectra of small measure which deliver positive solution of the interpolation problem in Bernstein spaces for every discrete sequence of knots.

Concordance crosscap number of a knot

AbstractWe define the concordance crosscap number γ c (K) of a knot K as the minimum crosscap number among all the knots concordant to K. The four-dimensional crosscap number γ*(K) is the minimum first Betti number of non-orientable surfaces smoothly embedded in the four-dimensional ball, bounding the knot K. Clearly, γ*(K) ≤ γ c (K). We construct two infinite sequences of knots for which γ*(K) < γ c (K). In particular, the knot 7 4 is one of the examples.

Hyperfinite knots via the CJKLS invariant in the thermodynamic limit

We set forth a definition of hyperfinite knots. Loosely speaking, these are limits of certain sequences of knots with increasing crossing number. These limits exist in appropriate closures of quotient spaces of knots. We give examples of hyperfinite knots. These examples stem from an application of the Thermodynamic Limit to the CJKLS invariant of knots.

A martingale approach to general Franklin systems

We prove unconditionality of general Franklin systems in $L^p(X)$, where $X$ is a UMD space and where the general Franklin system corresponds to a quasi-dyadic, weakly regular sequence of knots.

Efficient spatial and temporal representations of global ionosphere maps over Japan using B-spline wavelets

Wavelet expansion has been demonstrated to be suitable for the representation of spatial functions. Here we propose the so-called B-spline wavelets to represent spatial time-series of GPS-derived global ionosphere maps (GIMs) of the vertical total electron content (TEC) from the Earth’s surface to the mean altitudes of GPS satellites, over Japan. The scalar-valued B-spline wavelets can be defined in a two-dimensional, but not necessarily planar, domain. Generated by a sequence of knots, different degrees of B-splines can be implemented: degree 1 represents the Haar wavelet; degree 2, the linear B-spline wavelet, or degree 4, the cubic B-spline wavelet. A non-uniform version of these wavelets allows us to handle data on a bounded domain without any edge effects. B-splines are easily extended with great computational efficiency to domains of arbitrary dimensions, while preserving their properties. This generalization employs tensor products of B-splines, defined as linear superposition of products of univariate B-splines in different directions. The data and model may be identical at the locations of the data points if the number of wavelet coefficients is equal to the number of grid points. In addition, data compression is made efficient by eliminating the wavelet coefficients with negligible magnitudes, thereby reducing the observational noise. We applied the developed methodology to the representation of the spatial and temporal variations of GIM from an extremely dense GPS network, the GPS Earth Observation Network (GEONET) in Japan. Since the sampling of the TEC is registered regularly in time, we use a two-dimensional B-spline wavelet representation in space and a one-dimensional spline interpolation in time. Over the Japan region, the B-spline wavelet method can overcome the problem of bias for the spherical harmonic model at the boundary, caused by the non-compact support. The hierarchical decomposition not only allows an inexpensive calculation, but also separates visualisation at different levels of detail. Each level corresponds to a certain spatial frequency band, leading to a detection of structures and enhancement in the ionosphere at different resolutions.

General Franklin systems as bases in H1[0,1

By a general Franklin system corresponding to a dense sequence of knots ${\cal T}=(t_n, n \geq 0)$ in $[0,1]$ we mean a sequence of orthonormal piecewise linear functions with knots ${\cal T}$, that is, the $n$th function of the system has knots $t_0, \ld

Time‐sequenced Multi–Radio Frequency Observations of Cygnus X‐3 in Flare

Multifrequency observations from the VLA, VLBA and OVRO Millimeter Array of a major radio outburst of Cygnus X-3 in 2001 September are presented, measuring the evolution of the spectrum of the source over three decades in frequency, over a period of six days. Following the peak of the flare, as the intensity declines the high-frequency spectrum at frequency nu steepens from nu^{-0.4} to nu^{-0.6}, after which the spectral index remains at this latter terminal value; a trend previously observed but hitherto not satisfactorily explained. VLBA observations, for the first time, track over several days the expansion of a sequence of knots whose initial diameters are approximately 8 milliarcseconds. The light-crossing time within these plasmons is of the same order as the time-scale over which the spectrum is observed to evolve. We contend that properly accounting for light-travel time effects in and between plasmons which are initially optically thick, but which after expansion become optically thin, explains the key features of the spectral evolution, for example the observed timescale. Using the VLBA images, we have directly measured for the first time the proper motions of individual knots, analysis of which shows a two-sided jet whose axis is precessing. The best-fit jet speed is roughly beta = 0.63 and the precession period is about 5 days, significantly lower than fitted for a previous flare. Extrapolation of the positions of the knots measured by the VLBA back to zero-separation shows this to occur approximately 2.5 days after the detection of the rise in flux density of Cygnus X-3.

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree m if and only if it is a polynomial of degree m on every geometric sequence of knots. Here a sequence Kz with z 2 Z is called geometric if the knots Kz coincide outside a ball B, inside of which they satisfy Kz \B = z for all z and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in S 1 ◊ S 2 that can be distinguished by Z/2-invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over Z a universal Vassiliev invariant of degree 1 for knots in S 1 ◊S 2 .