Arbitrary Knots Research Articles

Periodic Orthonormal Spline Systems with Arbitrary Knots as Bases in $$\boldsymbol{H}^{\mathbf{1}}\boldsymbol{(\mathbb{T})}$$

Periodic Orthonormal Spline Systems with Arbitrary Knots as Bases in $$\boldsymbol{H}^{\mathbf{1}}\boldsymbol{(\mathbb{T})}$$

Spline representation and redundancies of one-dimensional ReLU neural network models

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to the model of continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give a recursive algorithm to transfer the parameter set determining the ReLU DNN into the parameter set of a CPL spline function. Using this representation, we show that after removing the well-known parameter redundancies of the ReLU DNN, which are caused by the positive scaling property, all remaining parameters are independent. Moreover, we show that the ReLU DNN with one, two or three hidden layers can represent CPL spline functions with K arbitrarily prescribed knots (breakpoints), where K is the number of real parameters determining the normalized ReLU DNN (up to the output layer parameters). Our findings are useful to fix a priori conditions on the ReLU DNN to achieve an output with prescribed breakpoints and function values.

Linear Phaseless Retrieval of Functions in Spline Spaces With Arbitrary Knots

We study phaseless retrieval in spline spaces generated by B-splines with arbitrary knots. For real spline spaces, we give a necessary and sufficient condition for a sequence of sampling points to admit a local phaseless retrieval of functions from its phaseless sampled values. We also study phaseless retrieval in complex spline spaces and illustrate that phase retrieval is impossible in this case. Nevertheless, we show that phaseless retrieval is possible. For a function in a complex spline space, no mater it is separable or not, we show that the square of its modulus is uniquely determined and can be recovered linearly from its sampled values at a well chosen sequence of sampling points. We give necessary and sufficient conditions for such sequences.

Almost Phaseless Sampling for Spline Spaces with Arbitrary Knots*

This article deals with almost phaseless sampling for spaces generated by B-splines with arbitrary knots. We provide a necessary and sufficient condition, which is a characterization of an almost phaseless sampling set, to admit phase recovery for almost all B-spline functions on an interval. Our results show that only N + 1 intensity measurements are sufficient to recover phase information for almost all functions in N dimensional B-spline function spaces with arbitrary knots.

A Definition of Two-Dimensional Schoenberg Type Operators

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity.

Geometry of Bend: Singular Lines and Defects in Twist-Bend Nematics.

We describe the geometry of bend distortions in liquid crystals and their fundamental degeneracies, which we call β lines; these represent a new class of linelike topological defect in twist-bend nematics. We present constructions for smecticlike textures containing screw and edge dislocations and also for vortexlike structures of double twist and Skyrmions. We analyze their local geometry and global structure, showing that their intersection with any surface is twice the Skyrmion number. Finally, we demonstrate how arbitrary knots and links can be created and describe them in terms of merons, giving a geometric perspective on the fractionalization of Skyrmions.

Refined Chern–Simons theory in genus two

Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus [Formula: see text] surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald [Formula: see text]-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117 opening a question that if it extends to any genus. Our construction is built upon a [Formula: see text]-deformation of the square of [Formula: see text]-6[Formula: see text] symbol of [Formula: see text], which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.

The Exact Form of the Second Moment of Third Degree Schoenberg Spline Operators

The article presents an explicit form for the second moment of cubic Schoenberg operators with arbitrary knots and in the particular case of equidistant knots. Applications are given for quantitative direct results in approximation and in construction of special classes of Schoenberg operators.

Evaluating the Jones polynomial with tensor networks.

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot is reduces to the partition function of a q-state anisotropic Potts model with complex interactions, which is defined on a planar signed graph that corresponds to the knot. For any integer q, we cast this partition function into tensor network form, which inherits the interaction graph structure of the Potts model instance, and employ fast tensor network contraction protocols to obtain the exact tensor trace and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace exactly, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.

Circuit complexity of knot states in Chern-Simons theory

We compute an upper bound on the circuit complexity of quantum states in 3d Chern-Simons theory corresponding to certain classes of knots. Specifically, we deal with states in the torus Hilbert space of Chern-Simons that are the knot complements on the 3-sphere of arbitrary torus knots. These can be constructed from the unknot state by using the Hilbert space representation of the S and T modular transformations of the torus as fundamental gates. The upper bound is saturated in the semiclassical limit of Chern-Simons theory. The results are then generalized for a family of multi-component links that are obtained by “Hopf-linking” different torus knots. We also use the braid word presentation of knots to discuss states on the punctured sphere Hilbert space associated with 2-bridge knots and links. The calculations present interesting number theoretic features related with continued fraction representations of rational numbers. In particular, we show that the minimization procedure defining the complexity naturally leads to regular continued fractions, allowing a geometric interpretation of the results in the Farey tesselation of the upper-half plane. Finally, we relate our discussion to the framework of path integral optimization by generalizing the original argument to non-trivial topologies.

Carbon nanotube knots

The structural and mechanical properties of carbon nanotube knots are investigated using molecular dynamics simulations. Using parametric representations of mathematical (harmonic) knots, a method is provided for calculating the initial atomic coordinates of carbon nanotubes in the shape of arbitrary knots for use in molecular dynamics simulations. A computational stress-strain testing scheme is implemented and applied to (5, 5) knotted carbon nanotubes to determine their tensile strength, plastic limit, and relative knot strength. Stress-strain curves are given for (5, 5) carbon nanotube stopper knots. It is determined that a carbon nanotube’s tensile strength is reduced to at most 1/3 of its original strength when tied into a knot. It is also shown that it is possible to form tight and stable carbon nanotube knots by subjecting the knots to stress beyond the plastic limit. In contrast, loose knots stabilized by noncovalent interactions are not dynamically stable and spontaneously untie. To help understand the stability of loose carbon nanotube knots, the relationship between bending strain energy and curvature is studied using carbon tori. Our study demonstrates the possibility to tie carbon nanotubes into various stable knots and provides a general framework for the study of other macromolecular knots relevant to potentially useful nanotechnology.

A Class of Birkhoff Type Interpolation and Applications

In this paper a class of Birkhoff type interpolation problem on arbitrary knots is studied. This class of Birkhoff interpolation is a generalization of a Birkhoff problem, which is included in one of the articles in the referenced list. Because the behavior of first and second order derivatives are more important than the higher order, therefore, for a basic function, a better approximation will be obtained than higher order derivatives. The low error value calculated for some functions using this class represents the relative importance of this class of Birkhoff interpolation problem. One of the uses of this class from the Birkhoff polynomials is to review and propose a quadrature formula for this class. One of the ways to reduce the error for interpolation polynomials of this class of Birkhoff interpolation problem is to select the Chebyshev polynomial zeros as nodal points, which can be used to solve numerical differential equations.

Quadruply-graded colored homology of knots

We conjecture the existence of four independent gradings in colored HOMFLYPT homology, and make qualitative predictions of various interesting structures and symmetries in the colored homology of arbitrary knots. We propose an explicit conjectural description for the rectangular colored homology of torus knots, and identify the new gradings in this context. While some of these structures have a natural interpretation in the physical realization of knot homologies based on counting supersymmetric configurations (BPS states, instantons, and vortices), others are completely new. They suggest new geometric and physical realizations of colored HOMFLYPT homology as the Hochschild homology of the category of branes in a Landau–Ginzburg B-model or, equivalently, in the mirror A-model. Supergroups and supermanifolds are surprisingly ubiquitous in all aspects of this work.

Orthogonal projectors onto spaces of periodic splines

The main result of this paper is a proof that for any integrable function f on the torus, any sequence of its orthogonal projections (P˜nf) onto periodic spline spaces with arbitrary knots Δ˜n and arbitrary polynomial degree converges to f almost everywhere with respect to the Lebesgue measure, provided the mesh diameter ∣Δ˜n∣ tends to zero. We also give a new and simpler proof of the fact that the operators P˜n are bounded on L∞ independently of the knots Δ˜n.

On the Jones Polynomial in the Solid Torus

We introduce an infinite class of elementary knots in the solid torus, together with general recursive and explicit formulas of the values of these knots under the generalized invariant of Jones polynomial to the solid torus. These values can be used as an infinite set of initial data for this invariant. We also introduce a procedure of resolving certain knots called spiral knots into these elementary knots. We show that our explicit formulas involve exactly \(n+1\) terms for an elementary knot with n crossings, which reduces the calculations needed to compute the invariant for spiral knots and arbitrary knots and links in the solid torus.

Vassiliev invariants for pretzel knots

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on [Formula: see text] parameters [Formula: see text]. These invariants are symmetric polynomials in [Formula: see text] whose degree coincide with their order. We also discuss their topological and integer-valued properties.

The SL(2, C) Casson Invariant for Knots and the Â-polynomial

AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

Local exponential splines with arbitrary knots

We construct local L-splines that have an arbitrary arrangement of knots and preserve the kernel of a linear differential operator L of order r with constant coefficients and real pairwise distinct roots of the characteristic polynomial.

Prony methods for recovery of structured functions

AbstractIn this survey, we describe the classical Prony method and whose relatives. We sketch a frequently used Prony–like method for equispaced sampled data, namely the ESPRIT method. The case of nonequispaced sampled data is discussed too. For the reconstruction of a sparse eigenfunction expansion, a generalized Prony method is presented. The Prony methods are applied to the recovery of structured functions (such as exponential sums and extended exponential sums) and of sparse vectors. The recovery of spline functions with arbitrary knots from Fourier data is also based on Prony methods. Finally, some numerical examples are given. (© 2014 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards matrix model representation of HOMFLY polynomials

We investigate possibilities of generalizing the TBEM eigenvalue matrix model, which represents the non-normalized colored HOMFLY polynomials for torus knots as averages of the corresponding characters. We look for a model of the same type, which is a usual Chern-Simons mixture of the Gaussian potential, typical for Hermitean models, and the sine Vandermonde factors, typical for the unitary ones. We mostly concentrate on the family of twist knots, which contains a single torus knot, the trefoil. It turns out that for the trefoil the TBEM measure is provided by an action of Laplace exponential on the Jones polynomial. This procedure can be applied to arbitrary knots and provides a TBEM-like integral representation for the N=2 case. However, beyond the torus family, both the measure and its lifting to larger N contain non-trivial corrections in \hbar=\log q. A possibility could be to absorb these corrections into a deformation of the Laplace evolution by higher Casimir and/or cut-and-join operators, in the spirit of Hurwitz tau-function approach to knot theory, but this remains a subject for future investigation.