A slope conjecture for links

2015 ◽  
Vol 24 (14) ◽  
pp. 1550077
Author(s):  
R. van der Veen
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Essential Surfaces ◽  
Torus Links ◽  
Boundary Slopes

The slope conjecture [S. Garoufalidis, The degree of a q-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011) 4–27] gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this paper, we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating or more generally adequate links. We also verify the conjecture for torus links.

scholarly journals The slope conjecture for Montesinos knots

2020 ◽  
Vol 31 (07) ◽  
pp. 2050056 ◽  
Author(s):  
Stavros Garoufalidis ◽  
Christine Ruey Shan Lee ◽  
Roland van der Veen
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Essential Surfaces ◽  
Sum Formula ◽  
Boundary Slopes

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

A result on the Slope conjectures for 3-string Montesinos knots

2018 ◽  
Vol 27 (13) ◽  
pp. 1842008
Author(s):  
Xudong Leng ◽  
Zhiqing Yang ◽  
Ximin Liu
Keyword(s):  
Integer Programming ◽  
Euler Characteristic ◽  
Jones Polynomial ◽  
Maximal Degree ◽  
Colored Jones Polynomial ◽  
Quadratic Integer Programming ◽  
Essential Surface ◽  
Essential Surfaces ◽  
Boundary Slope ◽  
Quadratic Integer

The Slope Conjecture and the Strong Slope Conjecture predict that the degree of the colored Jones polynomial of a knot is matched by the boundary slope and the Euler characteristic of some essential surfaces in the knot complement. By solving a problem of quadratic integer programming to find the maximal degree and using the Hatcher–Oertel edgepath system to find the corresponding essential surface, we verify the Slope Conjectures for a family of 3-string Montesinos knots satisfying certain conditions.

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 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.

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.

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