Related papers: The Morse-Novikov number of knots under connected …

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 $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two…

We define two new families of invariants for (3-manifold, graph) pairs which detect the unknot and are additive under connected sum of pairs and (-1/2)-additive under trivalent vertex sum of pairs. The first of these families is closely…

It is shown that for any locally knotted edge of a 3-connected graph in $S^3$, there is a ball that contains all of the local knots of that edge and is unique up to an isotopy setwise fixing the graph. This result is applied to the study of…

We study the AJ conjecture that relates the A-polynomial and the colored Jones polynomial of a knot in $S^3$. We confirm the AJ conjecture for $(r,2)$-cables of the $m$-twist knot, for all odd integers $r$ satisfying $\begin{cases}…

In the present note, we will show that there are infinitely many composite twisted torus knots.

The unknotting number of a positive braid with n strands and k intersections is known to be equal to (k-n+1)/2. We consider Lorenz knots (which are positive braids) and, using a different method, find their unknotting numbers in terms of…

In this paper, a generalized version of Morton's formula is proved. Using this formula, one can write down the colored Jones polynomials of cabling of an knot in terms of the colored Jones polynomials of the original knot.

We prove a formula for the involutive concordance invariants of the cabled knots in terms of that of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is…

This is a companion paper to earlier work of the author, which generalizes to an infinite family of $(2,2w+1)$-cabling of the figure eight knot ($|w|>3$) and proposes general formulas for the two-variable series invariant of the family of…

A multi-crossing (or n-crossing) is a singular point in a projection at which n strands cross so that each strand bisects the crossing. We generalize the classic result of Kauffman, Murasugi, and Thistlethwaite, which gives the upper bound…

For a knot $K,$ a slope $r$ is said to be characterizing if for no other knot $J$ does $r$-framed surgery along $J$ yield the same manifold as $r$-framed surgery on $K.$ Applying a condition of Baker and Motegi, we show that the knots…

Recent advances in Quantum Topology assign $q$-series to knots in at least three different ways. The $q$-series are given by generalized Nahm sums (i.e., special $q$-hypergeometric sums) and have unknown modular and asymptotic properties.…

We show that the crossing number of a satellite knot is at least 10^{-13} times the crossing number of its companion knot.

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a…

In this paper, we conjecture a connection between the $A$-polynomial of a knot in $\mathbb{S}^{3}$ and the hyperbolic volume of its exterior $\mathcal{M}_{K}$ : the knots with zero hyperbolic volume are exactly the knots with an…

This paper is part expository and part presentation of calculational results. The target space of the Kontsevich integral for knots is a space of diagrams; this space has various algebraic structures which are described here. These are…

Eisermann has shown that the Jones polynomial of a $n$-component ribbon link $L\subset S^3$ is divided by the Jones polynomial of the trivial $n$-component link. We improve this theorem by extending its range of application from links in…

We show that a band-connected sum of knots $K_0$ and $K_1$ along a band $b$ is equal to the connected sum $K_0\# K_1$ if and only if $b$ is a trivial band.

In this paper we use the connected sum operation on knots to show that there is a one-to-one relation between knots and numbers. In this relation prime knots are bijectively assigned with prime numbers such that the prime number 2…