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 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].

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

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 ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

scholarly journals THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS

2010 ◽  
Vol 19 (12) ◽  
pp. 1571-1595 ◽  
Author(s):  
STAVROS GAROUFALIDIS ◽  
XINYU SUN
Keyword(s):  
Difference Equation ◽  
Recursion Relation ◽  
Jones Polynomial ◽  
Minimal Order ◽  
Quantum Topology ◽  
And Topology ◽  
Jones Polynomials ◽  
Creative Telescoping ◽  
Colored Jones Polynomials ◽  
Telescoping Method

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial of a knot encodes the monic, linear, minimal order q-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to q = 1 is conjectured to be the better-known A-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the A-polynomial is harder to compute and already unknown for some knots with 12 crossings.

ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1001-1023 ◽  
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Closed Form ◽  
Explicit Expression ◽  
Computer Algebra ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Closed Form Formula

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

The generalized Alexander polynomial of periodic virtual links

2019 ◽  
Vol 28 (06) ◽  
pp. 1950042
Author(s):  
Joonoh Kim ◽  
Kyoung-Tark Kim ◽  
Mi Hwa Shin
Keyword(s):  
Alexander Polynomial ◽  
State Model ◽  
Quantum Algebras ◽  
Virtual Link ◽  
Linking Numbers ◽  
Virtual Links ◽  
Polynomial Formula

In this paper, we give several simple criteria to detect possible periods and linking numbers for a given virtual link. We investigate the behavior of the generalized Alexander polynomial [Formula: see text] of a periodic virtual link [Formula: see text] via its Yang–Baxter state model given in [L. H. Kauffman and D. E. Radford, Bi-oriented quantum algebras and a generalized Alexander polynomial for virtual links, in Diagrammatic Morphisms and Applications, Contemp. Math. 318 (2003) 113–140, arXiv:math/0112280v2 [math.GT] 31 Dec 2001].

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.

The coloured Jones function and Alexander polynomial for torus knots

1995 ◽  
Vol 117 (1) ◽  
pp. 129-135 ◽  
Author(s):  
H. R. Morton
Keyword(s):  
Power Series ◽  
Explicit Formula ◽  
Power Series Expansion ◽  
Irreducible Module ◽  
Alexander Polynomial ◽  
Torus Knots ◽  
Quantum Invariant ◽  
Torus Knot ◽  
Group Parameter ◽  
Jones Polynomials

AbstractIn [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion , where JK, k(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = eh is the quantum group parameter.In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

scholarly journals q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

2007 ◽  
Vol 16 (03) ◽  
pp. 267-332 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Braid Groups ◽  
Spin Networks ◽  
Spin Network ◽  
Topological Quantum ◽  
Jones Polynomials ◽  
Topological Quantum Computation ◽  
Bracket Polynomial ◽  
Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

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