Cable Knots Research Articles

Cable knots obtained from torus knots by twisting two strands

Cable knots obtained from torus knots by twisting two strands

Characterization of a spring pendulum phase-space trajectories.

We study the geometrical properties of phase-space trajectories (or orbits) of a spring pendulum as functions of the energy. Poincaré maps are used to describe the properties of the system. The points in the Poincaré maps of regular orbits (non-chaotic) cluster around separated segments or in chains of islands. Looking at how segments are formed, we conclude that the orbits are closely related to torus knots. Examining the toroidal and poloidal turns of the orbits, we introduce the definition of a rational parameter Ω, which is closely related to the concept of frequency used in the analysis of dynamical systems. Algorithms were developed to calculate Ω, and we found that this parameter naturally describes the orbits in terms of Farey sequences; also, calculations show that orbits with the same Ω have similar dynamics. Orbits corresponding to chains of islands are identified with cable knots that can be characterized using two parameters analogous to Ω. In some cases, non-trivial cable knots were found. With the analysis presented in this study, it is shown that Ω follows predictable distributions in the (z,Ω) space.

A Cable Knot and BPS-Series II

This is a companion paper to earlier work of the author, which generalizes to an infinite family of -cabling of the figure eight knot () and proposes general formulas for the two-variable series invariant of the family of the cable knots. The formulas provide an insight into the cabling operation. We verify the conjecture through explicit examples using the recursion method, which also provide a strong evidence for the q-holonomic property of the series invariant. This result paves a road for computation of the WRT invariant of a 3-manifold obtained from a Dehn surgery on the cable knots via a certain q-series. We also analyze and conjecture formulas for -cabling ().

Cable knots obtained by positively twisting torus knots

The twisted torus knot [Formula: see text] is a knot obtained from the torus knot [Formula: see text] by twisting [Formula: see text] adjacent stands fully [Formula: see text] times. If [Formula: see text] or [Formula: see text] is a multiple of [Formula: see text], then [Formula: see text] is known to be a cable of a nontrivial knot or a torus knot. Assuming that [Formula: see text] and [Formula: see text] is not a multiple of [Formula: see text], we determine the parameters [Formula: see text] for which the positively twisted torus knot [Formula: see text] is a cable knot.

A Note on Chirally Cosmetic Surgery on Cable Knots

AbstractWe show that a $(p,q)$-cable of a non-trivial knot K does not admit chirally cosmetic surgeries for $q\neq 2$, or $q=2$ with additional assumptions. In particular, we show that a $(p,q)$-cable of a non-trivial knot K does not admit chirally cosmetic surgeries as long as the set of JSJ pieces of the knot exterior does not contain the $(2,r)$-torus exterior for any r. We also show that an iterated torus knot other than the $(2,p)$-torus knot does not admit chirally cosmetic surgery.

Cable knots do not admit cosmetic surgeries

Two Dehn surgeries on a knot are called purely cosmetic if their surgered manifolds are homeomorphic as oriented manifolds. Gordon conjectured that nontrivial knots in [Formula: see text] do not admit purely cosmetic surgeries. In this paper, we confirm this conjecture for cable knots.

Representativity and waist of cable knots

We study the essential surfaces in the exterior of a cable knot to compute the representativity and waist of most cable knots. Our computation answers Ozawa’s question [5] about the relationship between the representativity and the waist of a knot in the negative.

Knotted optical vortices in exact solutions to Maxwell's equations

We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.

A cabling formula for the 𝜈⁺ invariant

We prove a cabling formula for the concordance invariant ν + \nu ^+ , defined by the author and Hom in a previous work. This gives rise to a simple and effective four-ball genus bound for many cable knots.

The strong AJ conjecture for cables of torus knots

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

Cabling in the Skyrme–Faddeev model

The Skyrme–Faddeev model is a three-dimensional nonlinear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the form of torus knots and links of these, however torus knots form only a small family of known knots. It is an open question whether any non-torus knot hopfions exist. In this paper we present a construction of knotted fields with the form of cable knots to which an energy minimization scheme can be applied. We find the first known hopfions which do not have the form of torus knots, but instead take the form of cable and hyperbolic knots.

Bordered Heegaard Floer homology and the tau-invariant of cable knots

We define a concordance invariant, epsilon(K), associated to the knot Floer complex of K, and give a formula for the Ozsv\'ath-Szab\'o concordance invariant tau of K_{p,q}, the (p,q)-cable of a knot K, in terms of p, q, tau(K), and epsilon(K). We also describe the behavior of epsilon under cabling, allowing one to compute tau of iterated cables. Various properties and applications of epsilon are also discussed.

Simplified Khovanov–Rozansky chain complexes of open 2-braids

Motivated by the works of Krasner (2009) [8] and Lobb [11], we simplify the Khovanov–Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a “long” 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.

Chern–Simons Invariants of Torus Links

We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We also derive a formula for cable knots. We use our results to test a recently proposed conjecture that relates HOMFLY and Kauffman invariants.

On Knot Floer Homology and Cabling: 2

We continue our study of the knot Floer homology invariants of cable knots. For large | n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p, pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsváth–Szabó concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and L-space surgeries.

Cable knots and infinite necklaces of knots

The group of an arbitrary companion knot is determined using the theory of braids. This seems to be a new result as far as the resulting group is concerned. The latter part of the paper considers infinite necklaces of knots, their groups and in special cases their Alexander power series.

Two-Generator Cable Knots are Tunnel One

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.

Two-generator cable knots are tunnel one

A two-generator cable knot exterior is a genus two handlebody with a single two-handle attached.

S- and T-matrices for the super U (1,1) WZW model application to surgery and 3-manifolds invariants based on the Alexander-Conway polynomial

We carry on (in a self-contained fashion) the study of the Alexander-Conway invariant from the quantum field theory point of view started earlier. We investigate for that purpose various aspects of WZW models on supergroups. We first discuss in details S- and T-matrices for the U(1,1) super WZW model and obtain, for the level k an integer, new finite-dimensional representations of the modular group. These have the remarkable property that some of the S-matrix elements are infinite (we show how to properly handle such divergences). Moreover, typical and atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated in some recent papers, is not consistent. Using our approach, multivariable Alexander invariants for links in S 3 can now be fully computed by surgery. Examples of torus and cable knots are discussed. Consistency with classical results provides independent checks of the solution of the U(1,1) WZW model. The main topological application of this work is the computation of Alexander invariants for 3-manifolds and more generally for links in 3-manifolds. Invariants of 3-manifolds themselves seem to depend trivially on the level k, but still contain interesting topological information. For Seifert manifolds for instance, they essentially coincide with the order (number of elements) of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties.

Fibered 2-knots and lens spaces

and pi>qi>0 (cf. [10], [13]). The punctured manifold M° means the space obtained from M by removing an open 3-ball. We consider fibered 2-knoΐs in S with fiber M°, which we call M°-fibered 2-knots. For example, a connected sum of 2-twist-spun knots of 2-bridge knots is such a fibered 2-knot. We shall determine possible M°-fibered 2-knots in S (or more generally, a homology 4-sρhere Σ) for all M. There are two reasons for selecting these fibered 2-knots: (1) Any 2-knot with Seifert manifold a connected sum of lens spaces is determined by its exterior [3], [4]. In particular, any fibered 2-knot with fiber a connected sum of lens spaces is determined by its monodromy (precisely, the class of its monodromy in the diίfeotopy group of the fiber). (2) The diffeotopy groups of all lens spaces are computed by Bonahon [1] and Hodgson-Rubinstein [7]. In section 3, we prove that any fibered 2-knot with fiber a punctured lens space L(p, q)° is the 2-twist-spun knot of K(p} q). In general, we show in section 4 that any M°-fibered 2-knot with cyclic monodromy is a cable knot about the 2-twist-spun knot of a 2-bridge knot. In section 5, we consider branched covering spaces of cable knots, and observe, for each odd r>l, the existence of a 2-knot which is the same fixed point set of many inequivalent semi-free ^-actions on S (cf. [2], [5], [14]). I would like to express my gratitude to Professor Akio Kawauchi for suggesting the problem, Professor Fujitsugu Hosokawa and Professor Yasutaka Nakanishi for leading me throughout.