Bing doubling and the colored Jones polynomial

Divisibility Property

Bing doubling is a satellite operation on links which replaces a knot component with a 2-component link in a certain way. In this paper we give a formula for the reduced colored Jones polynomial of a Bing double in terms of that of the companion. Using this formula we derive a divisibility property of the unified Witten–Reshetikhin–Turaev invariant of integral homology spheres obtained by ±1-surgery along Bing doubles of knots. This result is applied to the Witten–Reshetikhin–Turaev invariant and the Ohtsuki series of these integral homology spheres.

OHTSUKI INVARIANTS FOR INTEGRAL HOMOLOGY SPHERES AND HABIRO'S CYCLOTOMIC EXPANSION

10.1142/s021821650900680x ◽

2009 ◽

Vol 18 (01) ◽

pp. 21-31

Author(s):

TOSHIE TAKATA

Keyword(s):

Jones Polynomial ◽

Homology Sphere ◽

Integral Homology ◽

Borromean Rings

We give surgery formulas for the Ohtsuki invariants λ1, λ2and λ3of an integral homology sphere obtained by surgery from S3along a knot, related to Habiro's cyclotomic expansion of the colored Jones polynomial of the knot. As an application, we prove that the Ohtsuki invariants λ1, λ2and λ3separate integral homology 3-spheres obtained from S3by surgery along the Borromean rings.

On the universal sl2 invariant of Brunnian bottom tangles

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004112000503 ◽

2012 ◽

Vol 154 (1) ◽

pp. 127-143 ◽

Cited By ~ 2

Author(s):

SAKIE SUZUKI

Keyword(s):

Jones Polynomial ◽

Topological Properties ◽

Tensor Power ◽

Enveloping Algebra ◽

Divisibility Property ◽

Algebraic Properties ◽

Quantized Enveloping Algebra ◽

The Relationship ◽

Brunnian Links

AbstractThe universal sl2 invariant is an invariant of bottom tangles from which one can recover the colored Jones polynomial of links. We are interested in the relationship between topological properties of bottom tangles and algebraic properties of the universal sl2 invariant. A bottom tangle T is called Brunnian if every proper subtangle of T is trivial. In this paper, we prove that the universal sl2 invariant of n-component Brunnian bottom tangles takes values in a small subalgebra of the n-fold completed tensor power of the quantized enveloping algebra Uh(sl2). As an application, we give a divisibility property of the colored Jones polynomial of Brunnian links.

THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

10.1142/s0218216508006452 ◽

2008 ◽

Vol 17 (08) ◽

pp. 925-937

Author(s):

TOSHIFUMI TANAKA

Keyword(s):

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.

Stability properties of the colored Jones polynomial

10.1142/s0218216519500500 ◽

2019 ◽

Vol 28 (08) ◽

pp. 1950050

Author(s):

Christine Ruey Shan Lee

Keyword(s):

Power Series ◽

Jones Polynomial ◽

Topological Information ◽

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.

DIFFERENCE EQUATION OF THE COLORED JONES POLYNOMIAL FOR TORUS KNOT

International Journal of Mathematics ◽

10.1142/s0129167x04002582 ◽

2004 ◽

Vol 15 (09) ◽

pp. 959-965 ◽

Cited By ~ 22

Author(s):

KAZUHIRO HIKAMI

Keyword(s):

Difference Equation ◽

Jones Polynomial ◽

Order Difference ◽

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].

R-Matrix, the Colored Jones Polynomial, and the Kashaev Invariant

Volume Conjecture for Knots - SpringerBriefs in Mathematical Physics ◽

10.1007/978-981-13-1150-5_2 ◽

2018 ◽

pp. 11-26

Author(s):

Hitoshi Murakami ◽

Yoshiyuki Yokota

Keyword(s):

Jones Polynomial ◽

R Matrix ◽

Colored Jones Polynomial

Asymptotics of the colored Jones polynomial and the A-polynomial

Nuclear Physics B ◽

10.1016/j.nuclphysb.2007.03.022 ◽

2007 ◽

Vol 773 (3) ◽

pp. 184-202 ◽

Cited By ~ 5

Author(s):

Kazuhiro Hikami

Keyword(s):

Jones Polynomial ◽

Colored Jones Polynomial

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

10.1142/s0218216517410024 ◽

2017 ◽

Vol 26 (03) ◽

pp. 1741002 ◽

Cited By ~ 2

Author(s):

Mustafa Hajij

Keyword(s):

Natural Extension ◽

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.