Computer algebra and colored Jones polynomials

Compositio Mathematica, 2006

The colored Jones polynomial is a function J K : N −→ Z[t, t −1 ] associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of J K (n) are independent of n when n is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a Volume-ish Theorem for the colored Jones Polynomial.

Applied Mathematics and Computation, 2004

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Physics Letters B, 2015

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich (g + 1)parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY is fully and explicitly expressed through the Racah matrix of Uq(SUN), and looks related to a modular transformation of toric conformal block. Knot polynomials [1] are among the hottest topics in modern theory. They are supposed to summarize nicely representation theory of quantum algebras and modular properties of conformal blocks [2]-[6]. The result reported in the present letter, provides a spectacular illustration and support to this general expectation.

2011

Garoufalidis conjectured a relation between the boundary slopes of a knot and its colored Jones polynomials. According to the conjecture, certain boundary slopes are detected by the sequence of degrees of the colored Jones polynomials. We verify this conjecture for adequate knots, a class that vastly generalizes that of alternating knots.

Proceedings of the American Mathematical Society, 2016

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. We apply this to strengthen a theorem of Kalfagianni, Futer and Purcell on the fiberedness of adequate links.

Nuclear Physics B

The purpose of this paper is to introduce a closed form expression of the colored HOMFLY-PT polynomial for 3-braid knots. Indeed, we confine our interest to twisted generalized hybrid weaving knots which we denote hereafter asQ 3 (m 1 , −m 2 , n,). This family of knots generalizes the well-known class of weaving knots. Moreover, it contains an infinite family of quasi-alternating knots. Our main result is a closed form expression for the HOMFLY-PT polynomial ofQ 3 (m 1 , −m 2 , n,). Our approach is based on a modified version of the Reshitikhin-Turaev method. In addition, we compute the exact coefficients of the Jones polynomials and the Alexander polynomials of quasi-alternating knotsQ 3 (1, −1, n, ±1). For these homologically-thin knots, such coefficients are known to be the ranks of their Khovanov and link Floer homologies, respectively. We also show that the asymptotic behaviour of the coefficients of the Alexander polynomial is trapezoidal. On the other hand, we compute the [r]-colored HOMFLY-PT polynomials of quasi alternating knotsQ 3 (1, −1, n, ±1) for small values of r. Interestingly, the study of the determinants of certain twisted weaving knots leads to establish a connection with enumerative geometry related to m th Lucas numbers, denoted hereafter as L m,2n. At the end, we verify that the reformulated invariants from these quasi-alternating knot invariants satisfy Ooguri-Vafa conjecture and we express certain BPS integers in terms of hyper-geometric functions 2 F 1 a , b , c; z .

Discrete Mathematics & Theoretical Computer Science, 2009

Surveying the results of three recent papers and some currently ongoing research, we show how a generalization of Brylawski's tensor product formula to colored graphs may be used to compute the Jones polynomial of some fairly complicated knots and, in the future, even virtual knots.

Applied Mathematics and Computation, 2003

In this paper we apply computer algebra (MAPLE) techniques to calculate Alexander polynomial of (3,k)-Torus knots. For this purpose, a computer program was developed. When a positive integer k is given, the program calculates Alexander polynomials of (3,k)-Torus knots from Alexander matrix.

Eprint Arxiv Math 0301287, 2003

The paper computes the noncommutative A-ideal of the figure-eight knot, a noncommutative generalization of the A-polynomial. It is shown that if a knot has the same A-ideal as the figure-eight knot, then all colored Kauffman brackets are the same as those of the figure eight knot.

Journal of Physics A: Mathematical and Theoretical, 2017

Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site [1]. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, suggested in [2], and apply it to arborescent knots in Rolfsen table by developing a Feynman diagram technique, associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of nontrivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials.