Set Of Knots Research Articles

Natural classification of knots

The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. Here, we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages. In addition, we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot. These numbers include: crossing number; average three-dimensional writhe; number of topological domains; and the average relaxation value.

THE PENDULUM WEAVES ALL KNOTS AND LINKS

The behavior of a dissipative chaotic dynamical system is determined by the skeleton of its strange attractor, which consists of an uncountably infinite set of unstable periodic orbits. Each orbit is a topological knot, and the set is an infinite link. The types of knots and links supported by the system may be determined by collapsing the attractor along its local stable manifolds to form a template, a branched two manifold with boundary that supports the same set of knots and links as the original attractor. We show that the strange attractor of a chaotic, vertically forced physical pendulum can be collapsed to a template that supports all knots and links. Thus, one of the simplest and most well known dynamical systems is capable of the most complex behavior possible.

The loop expansion of the Kontsevich integral, the null-move and S-equivalence

The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Euler degree, and a geometric null-move on the set of knots. We explain the relation of the null-move to S-equivalence, and the relation to the Euler grading of the Kontsevich integral. The null-move leads in a natural way to the introduction of trivalent graphs with beads, and to a conjecture on a rational version of the Kontsevich integral, formulated by the second author and proven in Geom. Top 8 (2004) 115 (see also Kricker, preprint 2000, math/GT.0005284).

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 .

Sampling points on regular parametric curves with control of their distribution

We propose an iterative algorithm to generate a sequence of a prescribed number of points on a parametric curve with control of their distribution. Our algorithm depends on a free parameter which controls the achievement of a final distribution of points lying between two extreme cases: uniform arc length distribution and bending energy dependent distribution. This is obtained by computing a reparametrization function that interpolates the inverse of the arc length function at equidistant arguments. The proposed reparametrization function is a C 1 rational linear spline, therefore if the original curve is a NURBS then the reparametrized curve is also a NURBS of the same degree, but with an enlarged set of knots.

A surgery view of boundary links

A celebrated theorem of Kirby identifies the set of closed oriented connected 3-manifolds with the set of framed links in S 3 modulo two moves. We give a similar description for the set of knots (and more generally, boundary links) in homology 3-spheres. As an application, we define a noncommutative version of the Alexander polynomial of a boundary link. Our surgery view of boundary links is a key ingredient in a construction of a rational version of the Kontsevich integral, which is described in subsequent work.

Crossing change and exceptional Dehn surgery

Thurston’s hyperbolic Dehn surgery theorem [11], [12] asserts that if a knot in the 3-sphere 3 is hyperbolic (i.e., 3 − admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on yield hyperbolic 3-manifolds. By an exceptional surgery on a hyperbolic knot we mean a nontrivial Dehn surgery producing a non-hyperbolic manifold. Refer to [3], [6] for a survey on Dehn surgery on knots. We empirically know that ‘most’ knots are hyperbolic and ‘most’ hyperbolic knots have no exceptional surgeries. In this paper, we demonstrate the abundance of hyperbolic knots with no exceptional surgeries by showing that every knot is ‘close’ to infinitely many such hyperbolic knots in terms of crossing change. We regard that two knots are the same if they are isotopic in 3. For a knot in 3, let ( ) be the set of knots each of which is obtained by changing at most crossings in a diagram of .

STABLE EQUIVALENCE OF KNOTS ON SURFACES AND VIRTUAL KNOT COBORDISMS

We introduce an equivalence relation, called stable equivalence, on knot diagrams and closed generically immersed curves on surfaces. We give bijections between the set of abstract knots, the set of virtual knots, and the set of the stable equivalence classes of knot diagrams on surfaces. Using these bijections, we define concordance and link homology for virtual links. As an application, it is shown that Kauffman's example of a virtual knot diagram is not equivalent to a classical knot diagram.

On algorithms for ordinary least squares regression spline fitting: A comparative study

Regression spline smoothing is a popular approach for conducting nonparametric regression. An important issue associated with it is the choice of a "theoretically best" set of knots. Different statistical model selection methods, such as Akaike's information criterion and generalized cross-validation, have been applied to derive different "theoretically best" sets of knots. Typically these best knot sets are defined implicitly as the optimizers of some objective functions. Hence another equally important issue concerning regression spline smoothing is how to optimize such objective functions. In this article different numerical algorithms that are designed for carrying out such optimization problems are compared by means of a simulation study. Both the univariate and bivariate smoothing settings will be considered. Based on the simulation results, recommendations for choosing a suitable optimization algorithm under various settings will be provided.

Quantum Morphing and the Jones Polynomial

We will explore the experimental observation that on the set of knots with bounded crossing number, algebraically independent Vassiliev invariants become correlated, as noticed first by S. Willerton. We will see this through the value distribution of the Jones polynomial at roots of unit. As the degree of the roots of unit is getting larger, the higher order fluctuation is diminishing and a more organized shape will emerge from a rather random value distribution of the Jones polynomial. We call such a phenomenon “quantum morphing”. Evaluations of the Jones polynomial at roots of unity play a crucial role, for example in the volume conjecture.

Concordance and mutation

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n>1 there is an isomorphism phi from the concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of phi on C_1 is infinitely generated. Beyond this, little has been discovered about the pair (C_1,phi). In this paper we present a new approach to studying C_1 by introducing a group, M, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of phi, C_1-->M-->G_-. Our main result is that both maps have infinitely generated kernels. Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.

A data-adaptive knot selection scheme for fitting splines

A critical component of spline smoothing is the choice of knots, especially for curves with varying shapes and frequencies in its domain. We consider a two-stage knot selection scheme for adaptively fitting splines to data subject to noise. A potential set of knots is chosen based on information from certain wavelet decompositions with the intention of placing more points where the curve shows rapid changes. The final knot selection is then made based on statistical model selection ideas. We show that the proposed method is well suited for a variety of smoothing and noise filtering needs.

On $\pi$ -hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in \(S^3\) and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation \(K_1\approx K_2\) if \(M(K_1)\) is homeomorphic to \(M(K_2)\). It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related.

On the limiting power of the set of knots generated by 1+1- and 2+1-braids

We estimate from above the set of knots, $\Omega(n,\mu)$, generated by closure of n-string 1+1- and 2+1-dimensional braids of irreducible length $\mu$ ($\mu>>1$) in the limit n>>1.

Upper estimate of the cardinality of the set of knots generated by one-and two-dimensional braids

We give the upper estimate for the cardinality of the set Ω(n, μ) of knots generated by closed one- and two-dimensional braids with n generators of the irreducible length μ in the limit as n≫1 and μ≫1.

UNDERSTANDING THE KAUFFMAN BRACKET SKEIN MODULE

The Kauffman bracket skein module K(M) of a 3-manifold M is defined over formal power series in the variable h by letting A = eh/4. For a compact oriented surface F, it is shown that K(F×I) is a quantization of the [Formula: see text]-characters of the fundamental group of F corresponding to a geometrically defined Poisson bracket. Finite type invariants for unoriented knots and links are defined and obtained from topologically free Kauffman bracket modules. A structure theorem for K(M) is given in terms of the affine [Formula: see text]-characters of π1(M). It follows for compact M that K(M) can be generated as a module by cables on a finite set of knots. Moreover, if M contains no incompressible surfaces, the module is topologically finitely generated.

A finite set of generators for the Kauffman bracket skein algebra

If F is a compact orientable surface it is known that the Kauffman bracket skein module of \(F \times I\) has a multiplicative structure. Our central result is the construction of a finite set of knots which generate the module as an algebra. We can then define an integer valued invariant of compact orientable 3-manifolds which characterizes \(S^3\).

Modified Krylov acceleration for parallel environments

This paper considers a few variants of Krylov subspace techniques for solving linear systems on parallel computers. The goal of these variants is to avoid global dot-products which hamper parallelism in this class of methods. They are based on replacing the standard Euclidean inner product with a discrete inner product over polynomials. The set of knots for the discrete inner product is obtained by estimating eigenvalues of the coefficient matrix.

Modified (0, 2)-Interpolation on the Roots of Jacobi Polynomials. I (Explicit Formulae)

Let the set of knots $$ - 1 = x_{n + 1} < x_n < ... < x_1 < x_0 = 1 (n \geqq 1)$$ (n ≧ 1) be given on the interval [-1, 1]. Find a polynomial Qm(x) of minimal degree satisfying (0, 2)-interpolational conditions at the inner knots and boundary conditions at the endpoints, that is $$Q_m^{(s)} (x_i ) = y_i^{(s)} (s = 0,2) for i = 1,..., u_1$$ and $$Q_m^{(j)} (x_O ) = \alpha _O^{(j)} for j = 1,..., k$$ $$Q_m^{(j)} (x_{n + 1} ) = \alpha _{n + 1}^{(j)} for j = 0,..., l$$ where yi(s),αO(j), αn+1(j) are arbitrarily given real numbers, and k, l are arbitrary fixed non-negative integers. In this paper the existence and uniqueness of the polynomial Qm(x) is proved if the inner nodal points are the zeros of Jacobi polynomials Pn2k + 1, 2l − 1 (x) or Pn2k − 1, 2l + 1 (x). Explicit formulae for the fundamental polynomials of interpolation are also given.

Globally optimal periodic robot joint trajectories

This paper presents a new method for the planning of robot trajectories. The method presented assumes that joint-space knots have been generated from Cartesian knots by an inverse kinematics algorithm. The method is based on the globally optimal periodic interpolation scheme derived by Schoenberg, and thus is particularly suited for periodic robot motions. Of all possible periodic joint trajectories which pass through a specified set of knots, the trajectory derived in this paper is the ‘best’. The performance criterion used is the integral (over one period) of a combination of the square of the joint velocity and the square of the joint jerk.