相关论文: The non-commutative $A$-polynomial of twist knots

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation.…

We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we…

We conjecture formulae of the colored superpolynomials for a class of twist knots $K_p$ where p denotes the number of full twists. The validity of the formulae is checked by applying differentials and taking special limits. Using the…

We show that for a twist knot, the A-polynomial can be obtained from recurrences for the summand in Masbaum's formula of the colored Jones polynomial. Our result supports the AJ conjecture due to S.Garoufalidis.

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the…

In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The…

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ…

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.…

The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same…

We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, J_N(q = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion of…

The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the $\hat{A}$ polynomial), with a classical invariant, namely the defining polynomial $A$ of the $\psl$ character…

The paper shows the computation of the noncommutative generalization of the A-polynomial of the trefoil knot. The classical A-polynomial was introduced by Cooper, Culler, Gillet, Long and Shalen, and was generalized to the context of…

Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional…

The A-polynomial is a knot invariant related to the space of $SL_2(\mathbb{C})$ representations of the knot group. In this paper our interests lies in the logarithmic Gauss map of the A-polynomial. We develop a homological point of view on…

The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial…

We extend recent work by Howie, Mathews and Purcell to simplify the calculation of A-polynomials for any family of hyperbolic knots related by twisting. The main result follows from the observation that equations defining the deformation…

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…

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}…

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…

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…