相关论文: On the relation between the A-polynomial and the J…
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.…
We generalise the finite biquandle colouring invariant to a polynomial invariant based on labelling a knot diagram with a finite birack that reduces to the biquandle colouring invariant in that case. The polynomial is an invariant of a…
Habiro gave principal ideals of Z[q,q^{-1}] in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links…
This is an introduction to the Volume Conjecture and its generalizations for nonexperts. The Volume Conjecture states that a certain limit of the colored Jones polynomial of a knot would give the volume of its complement. If we deform the…
The volume conjecture and its generalization state that the series of certain evaluations of the colored Jones polynomials of a knot would grow exponentially and its growth rate would be related to the volume of a three-manifold obtained by…
The Kauffman-Vogel polynomials are three variable polynomial invariants of $4$-valent rigid vertex graphs. A one-variable specialization of the Kauffman-Vogel polynomials for unoriented $4$-valent rigid vertex graphs was given by using the…
We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open…
We prove an explicit cabling formula for the colored Jones polynomial. As an application we prove the volume conjecture for all zero volume knots and links, i.e. all knots and links that are obtained from the unknot by repeated cabling and…
It is still unknown whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. Tanaka shows that if an amphichiral knot is a symmetric union of the unknot with one twist region, then its Jones polynomial is…
The slope conjecture gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this note we propose a generalization of the slope…
We show that the equivariant and non-equivariant non-orientable 4-genus of p-periodic knots may differ, for any choice of p>1. Similar results have previously been obtained for the smooth 4-genus and non-orientable 3-genus of a periodic…
The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials.…
We study the head and tail of the colored Jones polynomial while focusing mainly on alternating links. Various ways to compute the colored Jones polynomial for a given link give rise to combinatorial identities for those power series. We…
We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link.…
This paper computes the Jones polynomial and the invariants obstructing cosmetic surgery which are derived from it for two infinite families of knots, proving they satisfy the Purely Cosmetic Surgery Conjecture. Both the method of…
We formulate a conjecture (already proven by A. Kricker) about the structure of Kontsevich integral of a knot. We describe its value in terms of the generating functions for the numbers of external edges attached to closed 3-valent…
The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this…
Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2…
It is shown that for knots with a sufficiently regular character variety the Dubois' torsion detects the A-polynomial of the knot. A global formula for the integral of the Dubois torsion is given. The formula looks like the heat kernel…
A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern--Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the…