Square numbers and polynomial invariants of achiral knots

We describe the leading coefficients of the Alexander and skein polynomial of alternating achiral knots. This work was supported by a JSPS postdoctoral grant.     

All frame-spun knots are slice

Frame-spun knots are constructed by spinning a knot of lower dimension about a framed submanifold of . We show that all frame-spun knots are slice (null-cobordant).

     

Periodic prime knots and topologically transitive flows on 3-manifolds

Suppose that φ is a nonsingular (fixed point free) C¹ flow on a smooth closed 3-dimensional manifold M with H₂(M)=0. Suppose that φ has a dense orbit. We show that there exists an open dense set N⊆M such that any knotted periodic orbit which intersects N is a nontrivial prime knot.     

Fibred knots and twisted Alexander invariants

We study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.

     

Evolutionary computation for optimal knots allocation in smoothing splines

In this paper, a novel methodology is presented for optimal placement and selections of knots, for approximating or fitting curves to data, using smoothing splines. It is well-known that the placement of the knots in smoothing spline approximation has an important and considerable effect on the behavior of the final approximation [1]. However, as pointed out in [2], although spline for approximation is well understood, the knot placement problem has not been dealt with adequately. In the specialized bibliography, several methodologies have been presented for selection and optimization of parameters within B-spline, using techniques based on selecting knots called dominant points, adaptive knots placement, by data selection process, optimal control over the knots, and recently, by using paradigms from computational intelligent, and Bayesian model for automatically determining knot placement in spline modeling. However, a common two-step knot selection strategy, frequently used in the bibliography, is an homogeneous distribution of the knots or equally spaced approach [3].     

On Legendrian knots and polynomial invariants

It is proved in this note that the analogues of the Bennequin inequality which provide an upper bound for the Bennequin invariant of a Legendrian knot in the standard contact three dimensional space in terms of the least degree in the framing variable of the HOMFLY and the Kauffman polynomials are not sharp. Furthermore, the relationships between these restrictions on the range of the Bennequin invariant are investigated, which leads to a new simple proof of the inequality involving the Kauffman polynomial.

     

Achirality of knots

A theoretic and diagrammatic relationship between knots and planar graphs has enabled us to establish necessary condition for achirality. It is shown that the regions, crossings and consequently the number of vertices, edges, and faces in the corresponding LR-graph are same and invariant. Establishment of new but pivotal moves such as R*-move, 2π-twist and π-twist enabled us to prove that the black regions can be changed into white regions via Reidemeister moves. Consequently, the equivalence of the companion graphs, necessary and sufficient conditions for achirality. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tunnel numbers of small knots do not go down under connected sum

Let and be two knots in and , the tunnel numbers of them. In this paper, we show that if both and are small, then . Moreover we show that for any small knots .

     

The Faddeev knots as stable solitons: Existence theorems

The problem of existence of knot-like solitons as the energy-minimizing configurations in the Faddeev model, topologically characterized by an Hopf invariant, Q, is considered. It is proved that, in the full space situation, there exists an infinite set S of integers so that for any m ∈ S, the Faddeev energy, E, has a minimizer among the class Q = m; in the bounded domain situation, the same existence theorem holds when S is the set of all integers. One of the important technical results is that E and Q satisfy the sublinear inequality E ≤ C|Q|3/4, where C ＞0 is a universal constant, which explains why knotted (clustered soliton) configurations are preferred over widely separated unknotted (multisoliton) configurations when |Q| is sufficiently large.     

Contact Dehn surgery, symplectic fillings, and Property P for knots

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Özbağcı–Stipsicz, a proof based on contact Dehn surgery is given of Eliashberg's concave filling theorem for contact 3-manifolds. The role of the theorem in the Kronheimer–Mrowka proof of Property P for nontrivial knots is sketched.     

The bracket semigroup of knots

The paper is devoted to a coding of links with marked point on an oriented component by means of regular bibracket structures, i.e., by some words in the alphabet (,),[,]. In this way we naturally obtain the semigroup of knots with concatenation as the semigroup operation, and with the equivalence classes modulo so-called “global relations” as elements. An important step in the construction of this semigroup is the coding of links with the help of so-calledd-diagrams. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 549–562, April, 2000.     

A mathematical model for enzymatic action on DNA knots and links

This paper deals with some studies pertaining te, nonprocessive recombinase viz. Topoisomerases III, IV. The mathematics of tangles is found to be very useful in studying topoisomerases and recombinases (processive and nonprocessive). It has been seen that the enzyme acts on the DNA if it is in a certain configuration. Electron micrographs of the enzyme-DNA complex show the enzyme as a blob with DNA looping out of it, but they are unable to determine the configuration of the DNA within the blob. By using knot theory and tangle calculus, the configuration of DNA within the enzyme blob as well as the enzyme action has been determined in saine cases.     

Knots of genus one or on the number of alternating knots of given genus

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

     

On $ \pi $-hyperbolic knots and branched coverings

We prove that, for any given , a -hyperbolic knot is determined by its 2-fold and n-fold cyclic branched coverings. We also prove that a -hyperbolic knot which is not determined by its m-fold and n-fold cyclic branched coverings, , must have genus . Received: December 14, 1998.     

Least squares approximation by splines with free knots

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process.The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.Dedicated to Professor Dr.-Ing. habil. Dr. h.c. Helmut Heinrich on the occasion of his 90th birthdayResearch of the second author was partly supported by Deutsche Forschungsgemeinschaft under grant Schm 968/2-1.     

Alternating knots satisfy strong property P

Suppose a manifold is produced by finite Dehn surgery on a non-torus alternating knot for which Seifert's algorithm produces a checkerboard surface. By demonstrating that it contains an essential lamination, we prove that such a manifold has as universal cover and, consequently, is irreducible and has infinite fundamental group. Together with previous work of Roberts, who proved this result in the case of alternating knots for which Seifert's algorithm does not produce a checkerboard surface, and Moser, who classified the manifolds produced by surgery on torus knots, this paper completes the proof that alternating knots satisfy Strong Property P. Received: May 20, 1998.     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.     

Knots with unknotting number one and Heegaard Floer homology

We use Heegaard Floer homology to give obstructions to unknotting a knot with a single crossing change. These restrictions are particularly useful in the case where the knot in question is alternating. As an example, we use them to classify all knots with crossing number less than or equal to nine and unknotting number equal to one. We also classify alternating knots with 10 crossings and unknotting number equal to one.     

On k -diameter of k -connected graphs

Let G be a k-connected simple graph with order n. The k-diameter, combining connectivity with diameter, of G is the minimum integer d _k (G) for which between any two vertices in G there are at least k internally vertex-disjoint paths of length at most d _k (G). For a fixed positive integer d, some conditions to insure d _k (G)⩽d are given in this paper. In particular, if d⩾3 and the sum of degrees of any s (s=2 or 3) nonadjacent vertices is at least n+(s−1)k+1−d, then d _k (G)⩽d. Furthermore, these conditions are sharp and the upper bound d of k-diameter is best possible. Supported by NNSF of China (19971086).