scholarly journals THE NONCOMMUTATIVE A-IDEAL OF A (2, 2p + 1)-TORUS KNOT DETERMINES ITS JONES POLYNOMIAL

2003 ◽  
Vol 12 (02) ◽  
pp. 187-201 ◽  
Author(s):  
RĂZVAN GELCA ◽  
JEREMY SAIN
Keyword(s):  
Jones Polynomial ◽  
Recursive Relation ◽  
Orthogonality Relation ◽  
Skein Modules ◽  
Kauffman Bracket ◽  
Torus Knot ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

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 colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.

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 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.

ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1001-1023 ◽  
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Closed Form ◽  
Explicit Expression ◽  
Computer Algebra ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Closed Form Formula

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

scholarly journals q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

2007 ◽  
Vol 16 (03) ◽  
pp. 267-332 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Braid Groups ◽  
Spin Networks ◽  
Spin Network ◽  
Topological Quantum ◽  
Jones Polynomials ◽  
Topological Quantum Computation ◽  
Bracket Polynomial ◽  
Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

DEGREES OF THE JONES POLYNOMIALS OF CERTAIN PRETZEL LINKS

2000 ◽  
Vol 09 (07) ◽  
pp. 907-916 ◽  
Author(s):  
MASAO HARA ◽  
SEI'ICHI TANI ◽  
MAKOTO YAMAMOTO
Keyword(s):  
Nonnegative Integer ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomials

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r _ deg VK=k for any nonnegative integer k, where c(K) is the crossing number of K and r _ deg VK is the reduced degree of the Jones polynomial VK of K.

Remarks on Chebyshev polynomials, Fibonacci polynomials and Kauffman bracket skein modules

2018 ◽  
Vol 27 (07) ◽  
pp. 1841007
Author(s):  
Robert Owczarek
Keyword(s):  
Chebyshev Polynomials ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Matrix Representations ◽  
Fibonacci Polynomials ◽  
Spin Structures ◽  
Skein Modules ◽  
Kauffman Bracket ◽  
Special Cases ◽  
Counting Solutions

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

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.

THE COMPLEX VOLUMES OF TWIST KNOTS VIA COLORED JONES POLYNOMIALS

2010 ◽  
Vol 19 (11) ◽  
pp. 1401-1421 ◽  
Author(s):  
JINSEOK CHO ◽  
JUN MURAKAMI
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Ideal Triangulation ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

For a hyperbolic knot, an ideal triangulation of the knot complement corresponding to the colored Jones polynomial was introduced by Thurston. Considering this triangulation of a twist knot, we find a function which gives the hyperbolicity equations and the complex volume of the knot complement, using Zickert's theory of the extended Bloch group and the complex volume. We also consider a formal approximation of the colored Jones polynomial. Following Ohnuki's theory of 2-bridge knots, we define another function which comes from the approximation. We show that this function is essentially the same as the previous function, and therefore it also gives the same hyperbolicity equations and the complex volume. Finally we compare this result with our previous one which dealt with Yokota theory, and, as an application to Yokota theory, present a refined formula of the complex volumes for any twist knots.

scholarly journals OPTIMISTIC LIMITS OF THE COLORED JONES POLYNOMIALS AND THE COMPLEX VOLUMES OF HYPERBOLIC LINKS

2016 ◽  
Vol 100 (3) ◽  
pp. 303-337 ◽  
Author(s):  
JINSEOK CHO
Keyword(s):  
Classical Limit ◽  
Mathematical Formulation ◽  
Jones Polynomial ◽  
Physical Method ◽  
Saddle Point Method ◽  
Colored Jones Polynomial ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials ◽  
Actual Limit

The optimistic limit is a mathematical formulation of the classical limit, which is a physical method to estimate the actual limit by using the saddle-point method of a certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. This modified version is easier to handle and more combinatorial than the original one. On the other hand, it is known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at a certain root of unity. This optimistic limit of the colored Jones polynomial was also formulated by the author and others, but it is very complicated and needs many unnatural assumptions. In this article, we suggest a modified optimistic limit of the colored Jones polynomial, following the idea of the modified optimistic limit of the Kashaev invariant, and show that it determines the complex volume of a hyperbolic link. Furthermore, we show that this optimistic limit coincides with the optimistic limit of the Kashaev invariant modulo $4{\it\pi}^{2}$. This new version is easier to handle and more combinatorial than the old version, and has many advantages over the modified optimistic limit of the Kashaev invariant. Because of these advantages, several applications have already appeared and more are in preparation.

scholarly journals AN EFFICIENT QUANTUM ALGORITHM FOR COLORED JONES POLYNOMIALS

2008 ◽  
Vol 06 (supp01) ◽  
pp. 773-778 ◽  
Author(s):  
MARIO RASETTI ◽  
SILVANO GARNERONE ◽  
ANNALISA MARZUOLI
Keyword(s):  
Jones Polynomial ◽  
Quantum Algorithm ◽  
Spin Network ◽  
Colored Jones Polynomial ◽  
Oriented Link ◽  
Root Of Unity ◽  
Jones Polynomials ◽  
Colored Jones Polynomials

We construct a quantum algorithm to approximate efficiently the colored Jones polynomial of the plat presentation of any oriented link L at a fixed root of unity q. The construction exploits the q-deformed spin network as computational background. The complexity of such algorithm is bounded above linearly by the number of crossings of the link, and polynomially by the number of link strands.

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