scholarly journals Kauffman–Jones polynomial of a curve on a surface

2017 ◽  
Vol 26 (11) ◽  
pp. 1750062
Author(s):  
Shinji Fukuhara ◽  
Yusuke Kuno
Keyword(s):  
Homotopy Class ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Laurent Polynomial ◽  
Computational Results ◽  
Oriented Surface ◽  
Variable Formula ◽  
Polynomial Formula

We introduce a Kauffman–Jones type polynomial [Formula: see text] for a curve [Formula: see text] on an oriented surface, whose endpoints are on the boundary of the surface. The polynomial [Formula: see text] is a Laurent polynomial in one variable [Formula: see text] and is an invariant of the homotopy class of [Formula: see text]. As an application, we obtain an estimate in terms of the span of [Formula: see text] for the minimum self-intersection number of a curve within its homotopy class. We then give a chord diagrammatic description of [Formula: see text] and show some computational results on the span of [Formula: see text].

scholarly journals GEOMETRY OF THE HOMOLOGY CURVE COMPLEX

2012 ◽  
Vol 04 (03) ◽  
pp. 335-359 ◽  
Author(s):  
INGRID IRMER
Keyword(s):  
Homotopy Class ◽  
Intersection Number ◽  
Simple Algorithm ◽  
Curve Complex ◽  
Oriented Surface ◽  
Immersed Surfaces ◽  
Minimal Genus ◽  
Closed Oriented Surface

Suppose S is a closed, oriented surface of genus at least two. This paper investigates the geometry of the homology multicurve complex, [Formula: see text], of S; a complex closely related to complexes studied by Bestvina–Bux–Margalit and Hatcher. A path in [Formula: see text] corresponds to a homotopy class of immersed surfaces in S × I. This observation is used to devise a simple algorithm for constructing quasi-geodesics connecting any two vertices in [Formula: see text], and for constructing minimal genus surfaces in S × I. It is proven that for g ≥ 3 the best possible bound on the distance between two vertices in [Formula: see text] depends linearly on their intersection number, in contrast to the logarithmic bound obtained in the complex of curves. For g ≥ 4 it is shown that [Formula: see text] is not δ-hyperbolic.

scholarly journals Cn-moves and the difference of Jones polynomials for links

2017 ◽  
Vol 26 (05) ◽  
pp. 1750029 ◽  
Author(s):  
Ryo Nikkuni
Keyword(s):  
Jones Polynomial ◽  
Laurent Polynomial ◽  
Link Invariant ◽  
Oriented Link ◽  
Jones Polynomials ◽  
The Difference ◽  
Polynomial Formula

The Jones polynomial [Formula: see text] for an oriented link [Formula: see text] is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer [Formula: see text], we show that: (1) the difference of Jones polynomials for two oriented links which are [Formula: see text]-equivalent is divisible by [Formula: see text], and (2) there exists a pair of two oriented knots which are [Formula: see text]-equivalent such that the difference of the Jones polynomials for them equals [Formula: see text].

scholarly journals The Alexander polynomial of a rational link

2019 ◽  
Vol 28 (03) ◽  
pp. 1950004
Author(s):  
Mark E. Kidwell ◽  
Kerry M. Luse
Keyword(s):  
Standard Form ◽  
Newton Polygon ◽  
Laurent Polynomial ◽  
Alexander Polynomial ◽  
Total Degree ◽  
Polynomial Formula ◽  
Positive Coefficients

We relate some terms on the boundary of the Newton polygon of the Alexander polynomial [Formula: see text] of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize [Formula: see text] to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then [Formula: see text]. If the standard form of the rational link has [Formula: see text] monochromatic twist sites, and the [Formula: see text]th monochromatic twist site has [Formula: see text] crossings, then [Formula: see text]. Our proof employs Kauffman’s clock moves and a lattice for the terms of [Formula: see text] in which the [Formula: see text]-power cannot decrease.

scholarly journals Square-free values of multivariate polynomials over function fields in linear sparse sets

2016 ◽  
Vol 13 (01) ◽  
pp. 77-108 ◽  
Author(s):  
Shai Rosenberg
Keyword(s):  
Function Fields ◽  
Multivariate Polynomials ◽  
Variable Formula ◽  
Integer Polynomials ◽  
Linear Condition ◽  
Polynomial Formula ◽  
Almost All

Let [Formula: see text] be a square-free polynomial where [Formula: see text] is a field of [Formula: see text] elements. We view [Formula: see text] as a polynomial in the variable [Formula: see text] with coefficients in the ring [Formula: see text]. We study square-free values of [Formula: see text] in sparse subsets of [Formula: see text] which are given by a linear condition. The motivation for our study is an analogue problem of representing square-free integers by integer polynomials, where it is conjectured that setting aside some simple exceptional cases, a square-free polynomial [Formula: see text] takes infinitely many square-free values. Let [Formula: see text] be co-prime to [Formula: see text], and let [Formula: see text]. A consequence of the main result we show is that if [Formula: see text] is sufficiently large with respect to [Formula: see text] and [Formula: see text], then there exist [Formula: see text] such that [Formula: see text] is square-free. Moreover, as [Formula: see text], the last is true for almost all [Formula: see text]. The main result shows that a similar result holds also for other cases. We then generalize the results to multivariate polynomials.

On lassos and the Jones polynomial of satellite knots

2016 ◽  
Vol 25 (02) ◽  
pp. 1650011
Author(s):  
Adrián Jiménez Pascual
Keyword(s):  
Jones Polynomial ◽  
Solid Torus ◽  
Alexander Polynomial ◽  
New Family ◽  
Jones Polynomials ◽  
Polynomial Formula ◽  
Satellite Knots

In this paper, I present a new family of knots in the solid torus called lassos, and their properties. Given a knot [Formula: see text] with Alexander polynomial [Formula: see text], I then use these lassos as patterns to construct families of satellite knots that have Alexander polynomial [Formula: see text] where [Formula: see text]. In particular, I prove that if [Formula: see text] these satellite knots have different Jones polynomials.

scholarly journals Frobenius algebras derived from the Kauffman bracket skein algebra

2016 ◽  
Vol 25 (04) ◽  
pp. 1650016 ◽  
Author(s):  
Charles Frohman ◽  
Nel Abdiel
Keyword(s):  
Fundamental Group ◽  
Central Extension ◽  
Function Field ◽  
Underlying Surface ◽  
Character Variety ◽  
Oriented Surface ◽  
Variable Formula ◽  
Kauffman Bracket ◽  
Frobenius Algebras ◽  
Character Ring

The Kauffman bracket skein algebra of a compact oriented surface when the variable [Formula: see text] in the Kauffman bracket is set equal to [Formula: see text], where [Formula: see text] is an odd counting number, is a central extension of the ring of [Formula: see text]-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

THE JONES POLYNOMIAL AND THE INTERSECTION NUMBERS OF TWISTED CYCLES ASSOCIATED WITH A SELBERG TYPE INTEGRAL

2011 ◽  
Vol 20 (03) ◽  
pp. 469-496 ◽  
Author(s):  
KATSUHISA MIMACHI
Keyword(s):  
Special Functions ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Homology Theory ◽  
Intersection Numbers ◽  
Type Integral ◽  
Jones Polynomials ◽  
Twisted Homology ◽  
Hypergeometric Type ◽  
Definition Of

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

Labels instead of coefficients: A label bracket ⋅ℒ which dominates the Jones polynomial 𝒳 ⋅, the Kuperberg bracket ⋅A2, and the normalized arrow polynomial 𝒲⋅

2020 ◽  
Vol 29 (02) ◽  
pp. 2040001
Author(s):  
Alyona A. Akimova ◽  
Vassily O. Manturov
Keyword(s):  
Jones Polynomial ◽  
Virtual Knots ◽  
Polynomial Formula

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial [Formula: see text], the Kuperberg bracket [Formula: see text], and the normalized arrow polynomial [Formula: see text].

scholarly journals A remarkable 20-crossing tangle

2017 ◽  
Vol 26 (14) ◽  
pp. 1750091
Author(s):  
Shalom Eliahou ◽  
Jean Fromentin
Keyword(s):  
Positive Integer ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Polynomial Formula ◽  
Bracket Polynomial

For any positive integer [Formula: see text], we exhibit a prime knot [Formula: see text] with [Formula: see text] crossings whose Jones polynomial [Formula: see text] is equal to [Formula: see text] modulo [Formula: see text]. Our construction rests on a certain [Formula: see text]-crossing prime tangle [Formula: see text] which is undetectable by the Kauffman bracket polynomial pair mod [Formula: see text].

scholarly journals IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL?

2006 ◽  
Vol 15 (08) ◽  
pp. 983-1000 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
THANG TQ LÊ
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Analytic Function ◽  
Jones Polynomial ◽  
Laurent Polynomial ◽  
Topological Quantum Field Theory ◽  
Quantum Field ◽  
Integer Coefficients ◽  
Topological Quantum ◽  
Proper Generalization

The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

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