Related papers: Properties of knots preserved by cabling

The $AJ$-conjecture for a knot $K \subset S^3$ relates the $A$-polynomial and the colored Jones polynomial of $K$. If a two-bridge knot $K$ satisfies the $AJ$-conjecture, we give sufficient conditions on $K$ for the $(r,2)$-cable knot $C$…

In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a conterexample for the conjecture which states that the height is additive under connected sum of knots. We…

We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic…

We show that if a knot has a minimal spanning surface that admits certain Gabai disks, then this knot has Property P. As one of the applications we extend and simplify a recent result of Menasco and Zhang that closed 3-braid knots have…

A slope $p/q$ is said to be characterizing for a knot $K$ if the homeomorphism type of the $p/q$-Dehn surgery along $K$ determines the knot up to isotopy. Extending previous work of Lackenby and McCoy on hyperbolic and torus knots…

A minimal knot is the intersection of a topologically embedded branched minimal disk in $\mathbb{R}^4$ $\mathbb{C}^2 $ with a small sphere centered at the branch point. When the lowest order terms in each coordinate component of the…

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided…

We study the behavior of the degree of the colored Jones polynomial and the boundary slopes of knots under the operation of cabling. We show that, under certain hypothesis on this degree, if a knot $K$ satisfies the Slope Conjecture then a…

Let M be $S^3$, $S^1\times S^2$, or a lens space L(p,q), and let k be a (1,1)-knot in M, i.e., a knot which is of 1-bridge with respect to a Heegaard torus. We show that if there is a closed meridionally incompressible surface in the…

We produce embeddings of knots in thin position that admit compressible thin levels. We also find the bridge number of tangle sums where each tangle is high distance.

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…

In this paper we give necessary and sufficient conditions for a knot type to admit non-loose Legendrian and transverse representatives in some overtwisted contact structure, classify all non-loose rational unknots in lens spaces, and…

Using a lattice model of polymers in a tube, we define one way to characterise different configurations of a given knot as either "local" or "non-local" and, for several ring polymer models, we provide both theoretical and numerical…

A method is introduced to isolate and measure the electrical transport properties of individual single-walled carbon nanotubes (SWNTs) aligned on an ST-cut quartz, from room temperature down to 2 K. The diameter and chirality of the…

We define combinatorial analogues of stable and unstable minimal surfaces in the setting of weighted pseudomanifolds. We prove that, under mild conditions, such combinatorial minimal surfaces always exist. We use a technique, adapted from…

For a polygonal knot K, it is shown that a tube of radius R(K), the polygonal thickness radius, is an embedded torus. Given a thick configuration K, perturbations of size r<R(K) define satellite structures, or local knotting. We explore…

In a lens space X of order r a knot K representing an element of the fundamental group pi_1 X = Z/rZ of order s <= r contains a connected orientable surface S properly embedded in its exterior X-N(K) such that the boundary of S intersects…

We show there exists a linear function w: N->N with the following property. Let K be a hyperbolic knot in a hyperbolic 3-manifold M admitting a non-longitudinal S^3 surgery. If K is put into thin position with respect to a strongly…

For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows…

We define homotopy-theoretic invariants of knots in prime 3-manifolds. Fix a knot J in a prime 3-manifold M. Call a knot K in M concordant to J if it cobounds a properly embedded annulus with J in MxI, and call K J-characteristic if there…