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1.
John A. Little 《Annali di Matematica Pura ed Applicata》1978,116(1):57-86
Summary Space curves may be classified under various kinds of deformation. The following six kinds of deformation have been of special
interest; namely first, second and third order homotopy and isotopy. (We say the deformation is k-th order if the first k
derivatives remain independent during the deformation.) The first order homotopy classification of space curves may be accomplished
using well-known methods of Whitney; there is only one class. The second and third order homotopy classification was done
by Feldman[1] and Little[6], respectively. The first order isotopy classification of space curves is knot theory; a subject of its own. The second order
isotopy classification has been done by W. F. Pohl (unpublished). Thus, aside from knot theory, the only remaining problem
is the third order isotopy problem. In this paper we give a partial answer. Our result is partial because we must restrict
the class of curves with which we are dealing; namely to curves with a ? twist ?. But it may well be that every curve does
have a twist, in which case our restricted class of curves would be all curves and the classification would be complete. In
addition we construct a curve of positive torsion with any preassigned self-linking number in any preassigned knot class;
a question raised by W. F. Pohl.
Entrata in Redazione il 4 settembre 1976. 相似文献
2.
Fumikazu Nagasato 《Topology and its Applications》2012,159(4):1059-1063
In this paper, applying Chebyshev polynomials we give a basic proof of the irreducibility over the complex number field of the defining polynomial of SL2(C)-character variety of twist knots in infinitely many cases. The irreducibility, combined with a result in the paper of M. Boileau, S. Boyer, A.W. Reid and S. Wang in 2010, shows the minimality of infinitely many twist knots for a partial order on the set of prime knots defined by using surjective group homomorphisms between knot groups. In Appendix B, we also give a straightforward proof of the result of Boileau et al. 相似文献
3.
Kondo and Sakai independently gave a characterization of Alexander polynomials for knots which are transformed into the trivial knot by a single crossing change. The first author gave a characterization of Alexander polynomials for knots which are transformed into the trefoil knot (and into the figure-eight knot) by a single crossing change. In this note, we will give a characterization of Alexander polynomials for knots which are transformed into the 10132 knot (and into the (5,2)-torus knot) by a single crossing change. Moreover, this method can be applied for knots with monic Alexander polynomials. 相似文献
4.
《Topology》1987,26(3):337-343
In this paper we show that any higher knot (n≥3) can be decomposed as a sum of irreducible knots and there is a finite upper bound on the number of summands. The case n=1 is due to Schubert [10]. A proof of this present case was published in [12] by Soninskii but subsequently Maeda showed that a crucial lemma was false [7]. The difficulty is to find a bound on decompositions of the knot group π1. This is achieved here by applying Dunwoody's work in [3]. This results in two theorems, 1.6 and 1.7, which are of some interest in their own right. The rest of the paper follows as in Sosinskii's paper.We are grateful to G.A. Swarup for pointing out the possibility of this line of attack and to A. Bartholomew for an idea used in the proof of Theorem 1.6. 相似文献
5.
H.A. Dye 《Topology and its Applications》2005,153(1):141-160
Kishino's knot is not detected by the fundamental group or the bracket polynomial. However, we can show that Kishino's knot is not equivalent to the unknot by applying either the 3-strand bracket polynomial or the surface bracket polynomial. In this paper, we construct two non-trivial virtual knot diagrams, KD and Km, that are not detected by the 1-strand or the 2-strand bracket polynomial. From these diagrams, we construct two infinite families of non-classical virtual knot diagrams that are not detected by the bracket polynomial. Additionally, these virtual knot diagrams are trivial as flats. 相似文献
6.
We note that the definition of R-functions depends on the choice of a certain surjection and pose the problem of the construction of a function of two variables
that is not an R-function for any choice of a surjective mapping. It is shown that the function x
1
x
2 − 1 possesses this property. We prove a theorem according to which, in the case of finite sets, every mapping is an R-mapping for a proper choice of a surjection. 相似文献
7.
Neil R. Hoffman 《Geometriae Dedicata》2014,173(1):299-308
Aitchison and Rubinstein showed there are two distinct ways to identify the faces of a pair of regular ideal dodecahedra and obtain a knot complement. This paper shows that these knot complements are the only knot complements that decompose into \(n\) regular ideal dodecahedra, providing a partial solution to a conjecture of Neumann and Reid. A corollary of the main theorem classifies the hyperbolic knot complements can be decomposed into regular ideal polyhedra. 相似文献
8.
Benoit Fresse 《Comptes Rendus Mathematique》2003,337(6):403-408
We prove that the bar construction of an E∞ algebra forms an E∞ algebra. To be more precise, we provide the bar construction of an algebra over the surjection operad with the structure of a Hopf algebra over the Barratt–Eccles operad. (The surjection operad and the Barratt–Eccles operad are classical E∞ operads.) To cite this article: B. Fresse, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
9.
Leonid Plachta 《Topology and its Applications》2007,154(15):2880-2887
Recently Stoimenow showed that for every knot K and any n∈N and u0?u(K) there is a prime knot Kn,uo which is n-equivalent to the knot K and has unknotting number u(Kn,uo) equal to u0. The similar result has been obtained for the 4-ball genus gs of a knot. Stoimenow also proved that any admissible value of the Tristram-Levine signature σξ can be realized by a knot with the given Vassiliev invariants of bounded order. In this paper, we show that for every knot K with genus g(K) and any n∈N and m?g(K) there exists a prime knot L which is n-equivalent to K and has genus g(L) equal to m. 相似文献
10.
A finite poset P(X,<) on a set X={
x
1,...,x
m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x
ij if and only if the angular region (regular n-gon) for x
i is contained in the region (regular n-gon) for x
j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415. 相似文献
11.
In this paper, we generalize two important results of Bagota and Móricz [1], and generalize our earlier results in [6] from one-variable to two-variable case. As special applications, we prove that the generalized jump of f(x, y) at some point (x 0, y 0) can be determined by the higher order mixed partial derivatives of the Abel-Poisson mean of double Fourier series and the higher order mixed partial derivatives of the Abel-Poisson means of the three conjugate double Fourier series. 相似文献
12.
This paper is concerned with α-convex operators on ordered Banach spaces. A surjection theorem for 1-convex operators in order intervals is established by means of the properties of cone and monotone iterative technique. It is assumed that 1-convex operator A is increasing and satisfies Ay−Ax?M(y−x) for θ?x?y?v0, where θ denotes the zero element and v0 is a constant. Moreover, we prove a fixed point theorem for -convex operators by using fixed point theorem of cone expansion. In the end, we apply the fixed point theorem to certain integral equations. 相似文献
13.
Valentin Gutev 《Topology and its Applications》2009,156(7):1172-1176
A continuous zero-selection f for the Vietoris hyperspace F(X) of the nonempty closed subsets of a space X is a Vietoris continuous map f:F(X)→X which assigns to every nonempty closed subset an isolated point of it. It is well known that a compact space X has a continuous zero-selection if and only if it is an ordinal space, or, equivalently, if X can be mapped onto an ordinal space by a continuous one-to-one surjection. In this paper, we prove that a compact space X has an upper semi-continuous set-valued zero-selection for its Vietoris hyperspace F(X) if and only if X can be mapped onto an ordinal space by a continuous finite-to-one surjection. 相似文献
14.
Let K be a knot in the 3-sphere S3 and D a disk in S3 meeting K transversely more than once in the interior. For nontriviality we assume that |D∩K|⩾2 over all isotopies of K in S3−∂D. Let KD,n (⊂S3) be a knot obtained from K by n twisting along the disk D. We prove that if K is a trivial knot and KD,n is a graph knot, then |n|⩽1 or K and D form a special pair which we call an “exceptional pair”. As a corollary, if (K,D) is not an exceptional pair, then by twisting unknot K more than once (in the positive or the negative direction) along the disk D, we always obtain a knot with positive Gromov volume. We will also show that there are infinitely many graph knots each of which is obtained from a trivial knot by twisting, but its companion knot cannot be obtained in such a manner. 相似文献
15.
Timothy H. McNicholl 《Mathematical Logic Quarterly》2003,49(3):219-229
We show that any partial order with a Σ3 enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co‐embeddable. 相似文献
16.
In recent years, several families of hyperbolic knots have been shown to have both volume and λ1 (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have
volume bounded by a generalized twist number. We show that for general knots, neither the twist number nor the generalized
twist number of a diagram can provide two-sided bounds on either the volume or λ1. We do so by studying the geometry of a family of hyperbolic knots that we call double coil knots, and finding two-sided bounds in terms of the knot diagrams on both the volume and on λ1. We also extend a result of Lackenby to show that a collection of double coil knot complements forms an expanding family iff their volume is bounded. 相似文献
17.
In this paper, we find the approximate solution of a second order nonlinear partial differential equation on a simple connected region inR 2. We transfer this problem to a new problem of second order nonlinear partial differential equation on a rectangle. Then, we transformed the later one to an equivalent optimization problem. Then we consider the optimization problem as a distributed parameter system with artificial controls. Finally, by using the theory of measure, we obtain the approximate solution of the original problem. In this paper also the global error inL 1 is controlled. 相似文献
18.
We consider functions which are subfunctions with respect to the differential operator
and are doubly periodic in ℂ. These functions play an important role in describing the asymptotic behavior of entire and
subharmonic functions of finite order [7, Ch. 3]. In studying their properties, we are led to problems concerning the uniqueness
of Martin functions and the critical value for the parameter ρ in the homogeneous boundary problem for the operatorL
ρ in a domain on the torus.
Supported in part by the National Science Foundation, Grant No. 9896337 and No. 9706408. 相似文献
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19.
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 41 in an arbitrary rectangular representation R = [rs] as a sum over all Young subdiagrams λ of R with surprisingly simple coefficients of the Z factors. Intriguingly, these coefficients are constructed from the quantum dimensions of symmetric representations of the groups SL(r) and SL(s) and restrict the summation to diagrams with no more than s rows and r columns. Moreover, the β-deformation to Macdonald dimensions yields polynomials with positive integer coefficients, which are plausible candidates for the role of superpolynomials for rectangular representations. Both the polynomiality and the positivity of the coefficients are nonobvious, nevertheless true. This generalizes the previously known formulas for symmetric representations to arbitrary rectangular representations. The differential expansion allows introducing additional gradings. For the trefoil knot 31, to which our results for the knot 41 are immediately extended, we obtain the so-called fourth grading of hyperpolynomials. The property of factorization in roots of unity is preserved even in the five-graded case. 相似文献
20.
In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system ${\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0}In this paper, we develop the basic concepts for a generalized Wiman–Valiron theory for Clifford algebra valued functions
that satisfy inside an n + 1-dimensional ball the higher dimensional Cauchy-Riemann system
\frac?f?x0 + ?i=1n ei\frac?f?xi=0{\frac{\partial f}{\partial x_0} + \sum_{i=1}^n e_i\frac{\partial f}{\partial x_i}=0} . These functions are called monogenic or Clifford holomorphic inside the ball. We introduce growth orders, the maximum term
and a generalization of the central index for monogenic Taylor series of finite convergence radius. Our goal is to establish
explicit relations between these entities in order to estimate the asymptotic growth behavior of a monogenic function in a
ball in terms of its Taylor coefficients. Furthermore, we exhibit a relation between the growth order of such a function f and the growth order of its partial derivatives. 相似文献