Related papers: On the Jones Polynomial
The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the…
We construct a bicomplex for the categorification of the colored Jones polynomial. This work is motivated by the problem suggested by Anna Beliakova and Stephan Wehrli who discussed the categorification of the colored Jones polynomial in…
Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
We give an explicit formula for the Jones polynomial of any rational link in terms of the denominators of the canonical continued fraction of the slope of the given rational link.
In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace.…
The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical…
The $A$-polynomial is conjectured to be obtained from the potential function of the colored Jones polynomial by elimination. The AJ conjecture also implies the relationship between the $A$-polynomial and the colored Jones polynomial. In…
The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given.…
It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs…
In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed. Orthogonal polynomials on the unit circle are not discussed.
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…
This is an expository paper on the subject of the title. It assumes basic scheme theory, commutative and homological algebra.
In this paper we give an introduction to the volume conjecture and its generalizations. Especially we discuss relations of the asymptotic behaviors of the colored Jones polynomials of a knot with different parameters to representations of…
An analytical expression for the third coefficient of the Jones Polynomial $P_J[\gamma,\, {\em e}^q]$ in the variable $q$ is reported. Applications of the result in Quantum Gravity are considered.
We give a topological formula of the loop expansion of the colored Jones polynomials by using identification of generic quantum sl2 representation with homological representations. This gives a direct topological proof of the…
We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce…
This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.
In the first of these two lectures, I use a comparison to symplectic Khovanov homology to motivate the idea that the Jones polynomial and Khovanov homology of knots can be defined by counting the solutions of certain elliptic partial…
We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call `Jones rational coincidences'. We provide moves on the continued fraction expansion of the associated rational which we prove do not…