Related papers: Recurrence relation for Jones polynomials
In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…
Using the Huynh and Le quantum determinant description of the colored Jones polynomial, we construct a new combinatorial description of the colored Jones polynomial in terms of walks along a braid. We then use this description to show that…
We derive a recurrence relation for the number of simple vertex-labelled bipartite graphs with given degrees of the vertices and use this result to obtain a new method for computing the growth function of the Artin monoid of type $A_{n-1}$…
A three term recurrence relation is derived for a basis consisting of polynomials multiplied by sines and cosines with large, but fixed frequencies. A numerical method for computing the coefficients of the three term recurrence relation is…
In this chapter are given necessary and sufficient conditions for the regularity of solutions of the functional equation appearing in the theory of classical orthogonal polynomials. In addition, we also present the functional Rodrigues…
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data…
We prove a stronger version of Jarden's Theorem for recurrence of powers of recursive functions
We obtain new recurrence relations, an explicit formula, and convolution identities for higher order geometric polynomials. These relations generalize known results for geometric polynomials, and lead to congruences for higher order…
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…
Motivated by the result that an `approximate' evaluation of the Jones polynomial of a braid at a $5^{th}$ root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP…
We provide a simple method to recognize classical orthogonal polynomials on lattices defined only by their coefficients of the three term recurrence relation.
We calculate Jones polynomials $V(H_r,t)$ for a family of alternating knots and links $H_r$ with arbitrarily many crossings $r$, by computing the Tutte polynomials $T(G_+(H_r),x,y)$ for the associated graphs $G_+(H_r)$ and evaluating these…
In this paper, a generalized version of Morton's formula is proved. Using this formula, one can write down the colored Jones polynomials of cabling of an knot in terms of the colored Jones polynomials of the original knot.
We prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…
We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem…
Entanglement of collections of filaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBC), which create an infinite periodic system whose global…
This paper describes a novel method to approximate the polynomial coefficients of regression functions, with particular interest on multi-dimensional classification. The derivation is simple, and offers a fast, robust classification…
We study the distribution of arithmetic invariants associated to Alexander polynomials for certain infinite families of links. The families of links we consider arise from braids on a fixed number of strings. We explore analogies with…
In this manuscript we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real…
This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our…