Related papers: Jones polynomials, volume, and essential knot surf…
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…
A simple geometric way is suggested to derive the Ward identities in the Chern-Simons theory, also known as quantum $A$- and $C$-polynomials for knots. In quasi-classical limit it is closely related to the well publicized augmentation…
Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials.…
State surfaces are spanning surfaces of links that are obtained from link diagrams guided by the combinatorics underlying Kauffman's construction of the Jones polynomial via state models. Geometric properties of such surfaces are often…
We show that the set of colored Jones polynomials and the set of generalized Alexander polynomials defined by Akutsu, Deguchi and Ohtsuki intersect non-trivially. Moreover it is shown that the intersection is (at least includes) the set of…
The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume…
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.
We show examples of knots with the same polynomial invariants and hyperbolic volumes, with variously coinciding 2-cable polynomials and colored Jones polynomials, which are not mutants.
This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones…
We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern-Simons invariants of the…
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones…
We construct 3D $\mathcal{N}=2$ abelian gauge theories on $\mathbb{S}^2 \times \mathbb{S}^1$ labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones…
In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I…
We extend the definition of the colored Jones polynomials to framed links and trivalent graphs in S^3 # k S^2 X S^1 using a state-sum formulation based on Turaev's shadows. Then, we prove that the natural extension of the Volume Conjecture…
In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984. The focus of our account will be recent glimmerings of understanding of the topological…
We show that colored Khovanov homology detects classes of essential surfaces as a direct analogue of the slope conjectures for the colored Jones polynomial. We do this by identifying certain generators of the colored Khovanov chain complex…
In the asymptotic expansion of the hyperbolic specification of the colored Jones polynomial of torus knots, we identify different geometric contributions, in particular Chern--Simons invaraint and Reidemeister torsion.
We prove the volume conjecture for any twist knots by using an equivalence relation, complex analysis, analytic continuation, and function of several complex variables on the basis of colored Jones polynomials.
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…
In this article, we survey the the recent literature surrounding the geometry of complex polynomials. Specific areas surveyed are i) Generalizations of the Gauss--Lucas Theorem, ii) Geometry of Polynomials Level Sets, and iii) Shape…