Related papers: The Strong Slope Conjecture and torus knots

We compute the Jones polynomial for a three-parameter family of links, the twisted torus links of the form $T((p,q),(2,s))$ where $p$ and $q$ are coprime and $s$ is nonzero. When $s = 2n$, these links are the twisted torus knots…

We reveal a relationship between the colored Jones polynomial and the A-polynomial for twist knots. We demonstrate that an asymptotics of the $N$-colored Jones polynomial in large $N$ gives the potential function, and that the A-polynomial…

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the colored Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov…

We propose a version of the volume conjecture that would relate a certain limit of the colored Jones polynomials of a knot to the volume function defined by a representation of the fundamental group of the knot complement to the special…

We show that for a torus knot the SL(2;C) Chern-Simons invariants and the SL(2;C) twisted Reidemeister torsions appear in an asymptotic expansion of the colored Jones polynomial. This suggests a generalization of the volume conjecture that…

We continue our study of the degree of the colored Jones polynomial under knot cabling started in "Knot Cabling and the Degree of the Colored Jones Polynomial" (arXiv:1501.01574). Under certain hypothesis on this degree, we determine how…

It is known that the colored Jones polynomials of a knot in the 3-dimensional sphere satisfy recursive relations, it is also known that these recursive relations come from recurrence polynomials which have been related, by the AJ…

We study relationships between the colored Jones polynomial and the A-polynomial of a knot. We establish for a large class of 2-bridge knots the AJ conjecture (of Garoufalidis) that relates the colored Jones polynomial and the A-polynomial.…

A technique to calculate the colored Jones polynomials of satellite knots, illustrated by the Whitehead doubles of knots, is presented. Then we prove the volume conjecture for Whitehead doubles of a family of torus knots and show some…

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 this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.

The signature function of a knot is a locally constant integer valued function with domain the unit circle. The jumps (i.e., the discontinuities) of the signature function can occur only at the roots of the Alexander polynomial on the unit…

The A-polynomial of a knot is defined in terms of SL(2,C) representations of the knot group, and encodes information about essential surfaces in the knot complement. In 2005, Dunfield-Garoufalidis and Boyer-Zhang proved that it detects the…

We formulate a conjecture about the structure of `upper lines' in the expansion of the colored Jones polynomial of a knot in powers of (q-1). The Melvin-Morton conjecture states that the bottom line in this expansion is equal to the inverse…

The set of isotopy classes of nontrivial torus knots $T(p,q)$ in $S^3$ is in bijection with the set of coprime integer pairs $(p,q)$ satisfying $|p|>q\geq 2$. We verify the AJ conjecture for the connected sums $T(p,q)\# T(a,b)$ when $p$ and…

We show that the triple-crossing number of any knot is greater or equal to twice its (canonical) genus and we show an even stronger bound in the case of links. As an application we show that this bound is strong enough to obtain the…

The extreme degrees of the colored Jones polynomial of any link are bounded in terms of concrete data from any link diagram. It is known that these bounds are sharp for semi-adequate diagrams. One of the goals of this paper is to show the…

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a $q$-state Potts model defined as a planar…

We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques…

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…