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 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.

THE COLORED JONES POLYNOMIALS OF 2-BRIDGE LINK AND HYPERBOLICITY EQUATIONS OF IT'S COMPLEMENTS

2005 ◽  
Vol 14 (06) ◽  
pp. 751-771 ◽  
Author(s):  
KOJI OHNUKI
Keyword(s):  
Jones Polynomial ◽  
Canonical Decomposition ◽  
Colored Jones Polynomial ◽  
Ideal Triangulation ◽  
Link Complement ◽  
Jones Polynomials ◽  
Colored Jones Polynomials ◽  
The Ideal

In this paper, we discuss the relation between the colored Jones polynomial of a 2-bridge link and the ideal triangulation of it's complement in S3. The aim of this paper is to describe the ideal triangulation of a 2-bridge link complement and to show that the hyperbolicity equations coincide with the equations obtained from the colored Jones polynomial of a 2-bridge link, and to compare this triangulation with the canonical decomposition of the 2-bridge link complement introduced by Sakuma and Weeks in [10].

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 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 Asymptotics of the colored Jones polynomial and the A-polynomial

2007 ◽  
Vol 773 (3) ◽  
pp. 184-202 ◽  
Author(s):  
Kazuhiro Hikami
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial

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.

scholarly journals On the Turaev–Viro endomorphism and the colored Jones polynomial

2013 ◽  
Vol 13 (1) ◽  
pp. 375-408
Author(s):  
Xuanting Cai ◽  
Patrick M Gilmer
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial

scholarly journals Bing doubling and the colored Jones polynomial

2014 ◽  
Vol 25 (08) ◽  
pp. 1450074
Author(s):  
S. Suzuki
Keyword(s):  
Jones Polynomial ◽  
Integral Homology ◽  
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.

scholarly journals On the colored Jones polynomial, sutured Floer homology, and knot Floer homology

2010 ◽  
Vol 223 (6) ◽  
pp. 2114-2165 ◽  
Author(s):  
J. Elisenda Grigsby ◽  
Stephan M. Wehrli
Keyword(s):  
Floer Homology ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Knot Floer Homology
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