On a Realization of Prime Tangles and Knots

The notion of a prime tangle is introduced by Kirby and Lickorish [7]. It is related deeply to the notion of a prime knot by the following result in [8]: summing together two prime tangles gives always a prime knot.The purpose of this paper is to exploit this above mentioned result of Lickorish in creating or detecting prime knots which satisfy certain properties. First, we shall express certain knots (two-bridge knots and Terasaka slice knots [14]) as a sum of a prime tangle and an untangle (the existence of such a sum is proven to every knot in [7] and is not unique) in a natural way (natural means here depending on certain specific geometrical characters of the class of knots). Second, every Alexander polynomial (or Conway polynomial) is shown to be realized by a prime algebraic knot (algebraic in the sense of Conway [3], Bonahon-Siebenmann [2]) which can be expressed as the sum of two prime algebraic tangles.

Explicit formulae for two-bridge knot polynomials

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700008004 ◽

2005 ◽

Vol 78 (2) ◽

pp. 149-166 ◽

Cited By ~ 5

Author(s):

Shinji Fukuhara

Keyword(s):

Normal Form ◽

Open Problem ◽

Alexander Polynomial ◽

Alexander Polynomials ◽

Elementary Number ◽

Conway Polynomial ◽

Coprime Integers ◽

Explicit Formulae ◽

Knots And Links

AbstractA two-bridge knot (or link) can be characterized by the so-called Schubert normal formKp, qwherepandqare positive coprime integers. Associated toKp, qthere are the Conway polynomial ▽kp, q(z)and the normalized Alexander polynomial Δkp, q(t). However, it has been open problem how ▽kp, q(z) and Δkp, q(t) are expressed in terms ofpandq. In this note, we will give explicit formulae for the Conway polynomials and the normalized Alexander polynomials in the case of two-bridge knots and links. This is done using elementary number theoretical functions inpandq.

Equivalence of Cables Of Mutants of Knots

Canadian Journal of Mathematics ◽

10.4153/cjm-1989-013-1 ◽

1989 ◽

Vol 41 (2) ◽

pp. 250-273 ◽

Cited By ~ 7

Author(s):

Józef H. Przytycki

Keyword(s):

Mirror Image ◽

Alexander Polynomial ◽

The Other ◽

Homfly Polynomial ◽

Similar Formula ◽

Conway Polynomial ◽

Other Hand ◽

Limited Possibility

There is the nice formula which links the Alexander polynomial of (m, k)-cable of a link with the Alexander polynomial of the link [5] [36] [38]. H. Morton and H. Short investigated whether a similar formula holds for the Jones-Conway (Homfly) polynomial and they found that it is very unlikely. Morton and Short made many calculations of the Jones-Conway polynomial of (2, q)-cables along knots (2 was chosen because of limited possibility of computers) and they get very interesting experimental material [24], [25]. In particular they found that using their method they were able to distinguish some Birman [4] and Lozano-Morton [22] examples (all which they tried) and the 942 knot (in the Rolfsen [37] notation) from its mirror image. On the other hand they were unable to distinguish the Conway knot and the Kinoshita-Terasaka knot.

On the Links–Gould invariant and the square of the Alexander polynomial

10.1142/s0218216516500061 ◽

2016 ◽

Vol 25 (02) ◽

pp. 1650006 ◽

Cited By ~ 1

Author(s):

Ben-Michael Kohli

Keyword(s):

Alexander Polynomial ◽

Braid Groups ◽

Direct Sum ◽

Conway Polynomial ◽

Exterior Powers

This paper gives a connection between well-chosen reductions of the Links–Gould invariants of oriented links and powers of the Alexander–Conway polynomial. This connection is obtained by showing the representations of the braid groups we derive the specialized Links–Gould polynomials from can be seen as exterior powers of a direct sum of Burau representations.

RECURSIVE CALCULATION FOR AN INVARIANT OF A RIBBON KNOT

10.1142/s0218216598000607 ◽

1998 ◽

Vol 07 (08) ◽

pp. 1093-1105 ◽

Cited By ~ 5

Author(s):

TAIZO KANENOBU

Keyword(s):

Finite Type ◽

Alexander Polynomial ◽

Second Derivative ◽

Conway Polynomial ◽

Type Invariant ◽

Recursive Calculation ◽

Second Degree

We give an algorithm for calculating the second degree coefficient of the Conway polynomial of a ribbon 1-knot. This naturally yields a recursive calculation for the second derivative at t = 1, Δ′′(1), of the normalized Alexander polynomial of a ribbon 2-knot in R4, which is the first nontrivial finite type invariant of a ribbon 2-knot defined by Habiro, Kanenobu, and Shima.

RIBBON LINK AND SEPARATE RIBBON LINK

10.1142/s0218216503002330 ◽

2003 ◽

Vol 12 (01) ◽

pp. 105-116 ◽

Cited By ~ 2

Author(s):

KAZUAKI KOBAYASHI ◽

KOUJI KODAMA ◽

TETSUO SHIBUYA

Keyword(s):

Higher Order ◽

Alexander Polynomial ◽

Conway Polynomial

We shall study several circles in the 3-sphere called a link which has "high" splitness properties. We offer several kind of those links and study relations among them. Alexander polynomial, Conway polynomial and Milnor μ and [Formula: see text] invariants did not work for those links as vanishing cause of high splitness. We use higher order elementary ideals to distinguish those links.

Linking Number of Singular Links and the Seifert Matrix

Canadian Mathematical Bulletin ◽

10.4153/cmb-2007-037-8 ◽

2007 ◽

Vol 50 (3) ◽

pp. 390-398 ◽

Cited By ~ 3

Author(s):

James J. Hebda ◽

Chun-Chung Hsieh ◽

Chichen M. Tsau

Keyword(s):

Alexander Polynomial ◽

Linking Number ◽

Half Integer ◽

Singular Link ◽

AbstractWe extend the notion of linking number of an ordinary link of two components to that of a singular link with transverse intersections, in which case the linking number is a half-integer. We then apply it to simplify the construction of the Seifert matrix, and therefore the Alexander polynomial, in a natural way.

SELF Ck-MOVE, QUASI SELF Ck-MOVE AND THE CONWAY POTENTIAL FUNCTION FOR LINKS

10.1142/s0218216504003500 ◽

2004 ◽

Vol 13 (07) ◽

pp. 877-893

Author(s):

TETSUO SHIBUYA ◽

AKIRA YASUHARA

Keyword(s):

Potential Function ◽

Necessary Conditions ◽

Alexander Polynomial ◽

Necessary Condition ◽

Conway Polynomial ◽

Link Homotopy

Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C1-equivalence and quasi self C2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self Ck-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self Ck-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

Application of a Simple Cold Stub to the Direct Observation of Frozen Hydrated Sand

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100071508 ◽

1973 ◽

Vol 31 ◽

pp. 212-213

Author(s):

John M. Wehrung ◽

Richard J. Harniman

Keyword(s):

United States ◽

Surface Water ◽

Direct Observation ◽

Controlled Study ◽

Clay Particles ◽

Organic Debris ◽

Rock Formations ◽

Water Tables ◽

Permeable Rock ◽

Natural Means

Water tables in aquifer regions of the southwest United States are dropping off at a rate which is greater than can be replaced by natural means. It is estimated that by 1985 wells will run dry in this region unless adequate artificial recharging can be accomplished. Recharging with surface water is limited by the plugging of permeable rock formations underground by clay particles and organic debris.A controlled study was initiated in which sand grains were used as the rock formation and water with known clay concentrations as the recharge media. The plugging mechanism was investigated by direct observation in the SEM of frozen hydrated sand samples from selected depths.

INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

Humanitarian actual problems of the humanities and education ◽

10.15507/2078-9823.045.019.201901.080-088 ◽

2019 ◽

Vol 19 (1) ◽

pp. 80-88

Author(s):

Nikolay S. Savkin

Keyword(s):

Social Relations ◽

Systematic Approach ◽

Human Life ◽

Laws Of Nature ◽

Arthur Schopenhauer ◽

Expected Effect ◽

Philosophy Of Life ◽

Active Subject ◽

Over Time ◽

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

Duality Myths

10.1093/oso/9780198809654.003.0009 ◽

2018 ◽

Author(s):

Elizabeth Schechter

Keyword(s):

Narrative Art ◽

Split Brain ◽

Dramatic Form ◽

This chapter addresses the intuitive fascination of the split-brain phenomenon. According to what I call the standard explanation, it is because we ordinarily assume that people are psychologically unified, while split-brain subjects are not psychologically unified, which suggests that we might not be unified either. I offer a different interpretation. One natural way of grappling with people’s failures to conform to various assumptions we make about them is to conceptualize them as having multiple minds. Such multiple-minds models take their most dramatic form in narrative art as duality myths. The split-brain cases grip people in part because the subjects strike them as living embodiments of such myths.