scholarly journals The strong AJ conjecture for cables of torus knots

2015 ◽  
Vol 24 (14) ◽  
pp. 1550072 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Jones Polynomial ◽  
Torus Knots ◽  
Colored Jones Polynomial ◽  
Pretzel Knots

The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cable knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cable knots over torus knots.

scholarly journals On the 2-head of the colored Jones polynomial for pretzel knots

2019 ◽  
Vol 70 (4) ◽  
pp. 1353-1370
Author(s):  
Paul Beirne
Keyword(s):  
Jones Polynomial ◽  
Infinite Family ◽  
Higher Order ◽  
Careful Examination ◽  
Colored Jones Polynomial ◽  
Stability Properties ◽  
Pretzel Knots

Abstract In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

scholarly journals Quantum modularity of partial theta series with periodic coefficients

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ankush Goswami ◽  
Robert Osburn
Keyword(s):  
Modular Form ◽  
Jones Polynomial ◽  
Theta Series ◽  
Periodic Coefficients ◽  
Torus Knots ◽  
Colored Jones Polynomial ◽  
Root Of Unity ◽  
The Family

Abstract We explicitly prove the quantum modularity of partial theta series with even or odd periodic coefficients. As an application, we show that the Kontsevich–Zagier series F t ⁢ ( q ) \mathscr{F}_{t}(q) which matches (at a root of unity) the colored Jones polynomial for the family of torus knots T ⁢ ( 3 , 2 t ) T(3,2^{t}) , t ≥ 2 t\geq 2 , is a weight 3 2 \frac{3}{2} quantum modular form. This generalizes Zagier’s result on the quantum modularity for the “strange” series F ⁢ ( q ) F(q) .

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.

DELTA-UNKNOTTING NUMBER FOR KNOTS

1998 ◽  
Vol 07 (05) ◽  
pp. 639-650 ◽  
Author(s):  
K. NAKAMURA ◽  
Y. NAKANISHI ◽  
Y. UCHIDA
Keyword(s):  
Torus Knots ◽  
Pretzel Knots ◽  
Minimum Number

The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.

scholarly journals Stability properties of the colored Jones polynomial

2019 ◽  
Vol 28 (08) ◽  
pp. 1950050
Author(s):  
Christine Ruey Shan Lee
Keyword(s):  
Power Series ◽  
Jones Polynomial ◽  
Topological Information ◽  
Colored Jones Polynomial ◽  
Stability Properties ◽  
Link Complement

It is known that the colored Jones polynomial of a [Formula: see text]-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the [Formula: see text]-adequate link complement. We show that a power series similar to the tail of the colored Jones polynomial for [Formula: see text]-adequate links can be defined for all links, and that it is trivial if and only if the link is non [Formula: see text]-adequate.

scholarly journals DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

2004 ◽  
Vol 15 (09) ◽  
pp. 959-965 ◽  
Author(s):  
KAZUHIRO HIKAMI
Keyword(s):  
Difference Equation ◽  
Jones Polynomial ◽  
Order Difference ◽  
Colored Jones Polynomial ◽  
First Order ◽  
Torus Knot ◽  
Second Order Difference Equation ◽  
Order Difference Equation ◽  
Hypergeometric Type ◽  
The Difference

We prove that the N-colored Jones polynomial for the torus knot [Formula: see text] satisfies the second order difference equation, which reduces to the first order difference equation for a case of [Formula: see text]. We show that the A-polynomial of the torus knot can be derived from the difference equation. Also constructed is a q-hypergeometric type expression of the colored Jones polynomial for [Formula: see text].

scholarly journals The colored Kauffman Skein relation and the head and tail of the colored Jones polynomial

2017 ◽  
Vol 26 (03) ◽  
pp. 1741002 ◽  
Author(s):  
Mustafa Hajij
Keyword(s):  
Natural Extension ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Alternating Links ◽  
Skein Relation

Using the colored Kauffman skein relation, we study the highest and lowest [Formula: see text] coefficients of the [Formula: see text] unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

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