scholarly journals Comparing nonorientable three genus and nonorientable four genus of torus knots

2020 ◽  
Vol 29 (03) ◽  
pp. 2050013
Author(s):  
Stanislav Jabuka ◽  
Cornelia A. Van Cott
Keyword(s):  
Congruence Class ◽  
Torus Knots ◽  
Torus Knot ◽  
The Difference ◽  
Fixed Congruence

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, let [Formula: see text] be any torus knot with [Formula: see text] even and [Formula: see text] odd. The difference between these two invariants on [Formula: see text] is at least [Formula: see text], where [Formula: see text] and [Formula: see text] and [Formula: see text]. Hence, the difference between the two invariants on torus knots [Formula: see text] grows arbitrarily large for any fixed odd [Formula: see text], as [Formula: see text] ranges over values of a fixed congruence class modulo [Formula: see text]. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot [Formula: see text] is [Formula: see text], and Kronheimer and Mrowka later proved that the orientable four genus of [Formula: see text] is also this same value.

Twisted torus knots T(mn + m + 1,mn + 1,mn + m + 2,−1) and T(n + 1,n,2n − 1,−1) are torus knots

2021 ◽  
pp. 2150016
Author(s):  
Sangyop Lee
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Braid Index

A twisted torus knot [Formula: see text] is a torus knot [Formula: see text] with [Formula: see text] adjacent strands twisted fully [Formula: see text] times. In this paper, we determine the braid index of the knot [Formula: see text] when the parameters [Formula: see text] satisfy [Formula: see text]. If the last parameter [Formula: see text] additionally satisfies [Formula: see text], then we also determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot.

TORUS KNOTS EMBODYING CURVATURE AND TORSION IN AN OTHERWISE FEATURELESS CONTINUUM

2011 ◽  
Vol 20 (12) ◽  
pp. 1723-1739 ◽  
Author(s):  
J. S. AVRIN
Keyword(s):  
Elementary Particle ◽  
General Relativity ◽  
Geometric Model ◽  
The Other ◽  
Particle Model ◽  
Torus Knots ◽  
Torus Knot ◽  
The Subject ◽  
Curvature And Torsion ◽  
The Relationship

The subject is a localized disturbance in the form of a torus knot of an otherwise featureless continuum. The knot's topologically quantized, self-sustaining nature emerges in an elementary, straightforward way on the basis of a simple geometric model, one that constrains the differential geometric basis it otherwise shares with General Relativity (GR). Two approaches are employed to generate the knot's solitonic nature, one emphasizing basic differential geometry and the other based on a Lagrangian. The relationship to GR is also examined, especially in terms of the formulation of an energy density for the Lagrangian. The emergent knot formalism is used to derive estimates of some measurable quantities for a certain elementary particle model documented in previous publications. Also emerging is the compatibility of the torus knot formalism and, by extension, that of the cited particle model, with general relativity as well as with the Dirac theoretic notion of antiparticles.

scholarly journals RIBBONLENGTH OF TORUS KNOTS

2008 ◽  
Vol 17 (01) ◽  
pp. 13-23 ◽  
Author(s):  
BROOKE KENNEDY ◽  
THOMAS W. MATTMAN ◽  
ROBERTO RAYA ◽  
DAN TATING
Keyword(s):  
Crossing Number ◽  
Regular Polygons ◽  
Torus Knots ◽  
Torus Knot

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1) cot (π/(2q + 1)) (respectively, 2q cot (π/(2q + 1))). Using these calculations, we provide the bounds c1 ≤ 2/π and c2 ≥ 5/3 cot π/5 for the constants c1 and c2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c1 C(K) ≤ R(K) ≤ c2 C(K).

scholarly journals Dichromatic polynomial for graph of a (2,n)-torus knot

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Abdulgani Şahin
Keyword(s):  
Tutte Polynomial ◽  
Signed Graph ◽  
Torus Knots ◽  
Torus Knot ◽  
Tutte Polynomials ◽  
The Relationship ◽  
Dichromatic Polynomial

AbstractIn this study, we introduce the relationship between the Tutte polynomials and dichromatic polynomials of (2,n)-torus knots. For this aim, firstly we obtain the signed graph of a (2,n)-torus knot, marked with {+} signs, via the regular diagram of its. Whereupon, we compute the Tutte polynomial for this graph and find a generalization through these calculations. Finally we obtain dichromatic polynomial lying under the unmarked states of the signed graph of the (2,n)-torus knots by the generalization.

Positively twisted torus knots which are torus knots

2019 ◽  
Vol 28 (03) ◽  
pp. 1950023 ◽  
Author(s):  
Sangyop Lee
Keyword(s):  
Torus Knots ◽  
Torus Knot

A twisted torus knot [Formula: see text] is obtained from a torus knot [Formula: see text] by twisting [Formula: see text] adjacent strands of [Formula: see text] fully [Formula: see text] times. In this paper, we determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot with [Formula: see text].

scholarly journals FACTORING FAMILIES OF POSITIVE KNOTS ON LORENZ-LIKE TEMPLATES

2008 ◽  
Vol 17 (10) ◽  
pp. 1175-1187 ◽  
Author(s):  
MICHAEL C. SULLIVAN
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Closed Orbits ◽  
Prime Factors

We show that for m and n positive, composite closed orbits realized on the Lorenz-like template L(m, n) have two prime factors, each a torus knot; and that composite closed orbits on L(-1, -1) have either two for three prime factors, two of which are torus knots.

scholarly journals Additional cases of positive twisted torus knots

2017 ◽  
Vol 26 (12) ◽  
pp. 1750078
Author(s):  
Evan Amoranto ◽  
Brandy Doleshal ◽  
Matt Rathbun
Keyword(s):  
Sufficient Conditions ◽  
Parameter Family ◽  
Surface Slope ◽  
Torus Knots ◽  
Torus Knot ◽  
Genus 2 ◽  
Heegaard Surface

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in [Formula: see text], the genus 2 Heegaard surface for [Formula: see text]. Primitive/primitive and primitive/Seifert knots lie in [Formula: see text] in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in [Formula: see text]. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in [Formula: see text].

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 FUNDAMENTAL BIQUANDLES OF RIBBON 2-KNOTS AND RIBBON TORUS-KNOTS WITH ISOMORPHIC FUNDAMENTAL QUANDLES

2014 ◽  
Vol 23 (01) ◽  
pp. 1450001 ◽  
Author(s):  
SOSUKE ASHIHARA
Keyword(s):  
Torus Knots ◽  
Torus Knot

The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

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.

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