ON DOUBLE-TORUS KNOTS (I)

1999 ◽  
Vol 08 (08) ◽  
pp. 1009-1048 ◽  
Author(s):  
PETER HILL
Keyword(s):  
Alexander Polynomial ◽  
Torus Knots ◽  
Torus Knot ◽  
Genus Two ◽  
Double Torus ◽  
Heegaard Surface

A double-torus knot is knot embedded in a genus two Heegaard surface [Formula: see text] in S3. After giving a notation for these knots, we consider double-torus knots L such that [Formula: see text] is not connected, and give a criterion for such knots to be non-trivial. Various new types of non-trivial knots with trivial Alexander polynomial are found.

ON DOUBLE-TORUS KNOTS (II)

2000 ◽  
Vol 09 (05) ◽  
pp. 617-667 ◽  
Author(s):  
PETER HILL ◽  
KUNIO MURASUGI
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Genus Two ◽  
Double Torus ◽  
Heegaard Surface

A double-torus knot is a knot embedded in a genus two Heegaard surface [Formula: see text] in S3. We consider double-torus knots L such that [Formula: see text] is connected, and consider fibred knots in various classes.

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].

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.

Trivial Double-Torus Knot

2003 ◽  
Vol 12 (05) ◽  
pp. 579-588
Author(s):  
Chuichiro Hayashi
Keyword(s):  
Heegaard Splitting ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Torus Knot ◽  
Genus Two ◽  
Double Torus ◽  
Necessary And Sufficient

A knot K in a closed connected orientable 3-manifold M is called a double-torus knot, if it is in a genus two Heegaard splitting surface H of M. We give a necessary and sufficient condition for a double-torus knot to be the trivial knot in words of meridian disks of genus two handlebodies obtained by splitting M along H.

SATELLITE DOUBLE TORUS KNOTS

2001 ◽  
Vol 10 (01) ◽  
pp. 133-142 ◽  
Author(s):  
MAKOTO OZAWA
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Double Torus

We characterize satellite double torus knots. Especially, if a satellite double torus knot is not a cable knot, then it has a torus knot companion. This answers Question 12 (a) raised by Hill and Murasugi in [4].

scholarly journals ASYMPTOTIC BEHAVIORS OF THE COLORED JONES POLYNOMIALS OF A TORUS KNOT

2004 ◽  
Vol 15 (06) ◽  
pp. 547-555 ◽  
Author(s):  
HITOSHI MURAKAMI
Keyword(s):  
Alexander Polynomial ◽  
The Other ◽  
Torus Knots ◽  
Asymptotic Behaviors ◽  
Torus Knot ◽  
Other Hand ◽  
Jones Polynomials ◽  
Colored Jones Polynomials ◽  
Chern Simons

We study the asymptotic behaviors of the colored Jones polynomials of torus knots. Contrary to the works by R. Kashaev, O. Tirkkonen, Y. Yokota, and the author, they do not seem to give the volumes or the Chern–Simons invariants of the three-manifolds obtained by Dehn surgeries. On the other hand it is proved that in some cases the limits give the inverse of the Alexander polynomial.

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.

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