scholarly journals Recurrence Relations and Asymptotics of Colored Jones Polynomials

2021 ◽  
Vol 42 (11) ◽  
pp. 2580-2595
Author(s):  
A. I. Aptekarev ◽  
T. V. Dudnikova ◽  
D. N. Tulyakov
Keyword(s):  
Difference Equations ◽  
Recurrence Relations ◽  
High Order ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

Abstract We consider $$q$$-difference equations for colored Jones polynomials. These sequences of polynomials are invariants for the knots and their asymptotics plays an important role in the famous volume conjecture for the complement of the knot to the $$3$$d sphere. We give an introduction to the theory of hyperbolic volume of the knots complements and study the asymptotics of the solutions of $$q$$-recurrence relations of high order.

THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

2008 ◽  
Vol 17 (08) ◽  
pp. 925-937
Author(s):  
TOSHIFUMI TANAKA
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Skein Theory ◽  
Colored Jones Polynomials

We give formulas for the N-colored Jones polynomials of doubles of knots by using skein theory. As a corollary, we show that if the volume conjecture for untwisted positive (or negative) doubles of knots is true, then the colored Jones polynomial detects the unknot.

scholarly journals COLORED JONES POLYNOMIALS WITH POLYNOMIAL GROWTH

2008 ◽  
Vol 10 (supp01) ◽  
pp. 815-834 ◽  
Author(s):  
KAZUHIRO HIKAMI ◽  
HITOSHI MURAKAMI
Keyword(s):  
Irreducible Representation ◽  
Complex Number ◽  
Polynomial Growth ◽  
Jones Polynomial ◽  
The Other ◽  
Absolute Value ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Dimensional Irreducible Representation ◽  
Colored Jones Polynomials

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;ℂ) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near [Formula: see text]. On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp c. In this paper we study cases where it grows polynomially.

scholarly journals THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

2010 ◽  
Vol 19 (12) ◽  
pp. 1571-1595 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
XINYU SUN
Keyword(s):  
Difference Equation ◽  
Recursion Relation ◽  
Jones Polynomial ◽  
Minimal Order ◽  
Quantum Topology ◽  
And Topology ◽  
Jones Polynomials ◽  
Creative Telescoping ◽  
Colored Jones Polynomials ◽  
Telescoping Method

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

scholarly journals The colored Jones polynomials and the simplicial volume of a knot

2001 ◽  
Vol 186 (1) ◽  
pp. 85-104 ◽  
Author(s):  
Hitoshi Murakami ◽  
Jun Murakami
Keyword(s):  
Simplicial Volume ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

scholarly journals The Hermite polynomials and the Bessel functions from a general point of view

2003 ◽  
Vol 2003 (57) ◽  
pp. 3633-3642 ◽  
Author(s):  
G. Dattoli ◽  
H. M. Srivastava ◽  
D. Sacchetti
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Bessel Functions ◽  
Hermite Polynomials ◽  
Point Of View ◽  
General Point ◽  
Ordinary Case

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

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