scholarly journals Non-commutative trigonometry and the A-polynomial of the trefoil knot

2002 ◽  
Vol 133 (2) ◽  
pp. 311-323 ◽  
Author(s):  
RĂZVAN GELCA
Keyword(s):  
Character Variety ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Trefoil Knot ◽  
Sum Formula ◽  
Left And Right ◽  
Trefoil Knots ◽  
The Relationship ◽  
The Ideal

The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character variety of the fundamental group, which was explained in [2]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.

THE CHARACTER VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

2012 ◽  
Vol 23 (01) ◽  
pp. 1250015 ◽  
Author(s):  
KHALED QAZAQZEH
Keyword(s):  
Character Variety ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
One Relator Groups

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link Lp/q of two components modulo the (A + 1)-torsion.

scholarly journals The Kauffman bracket skein module of the handlebody of genus 2 via braids

2019 ◽  
Vol 28 (13) ◽  
pp. 1940020
Author(s):  
Ioannis Diamantis
Keyword(s):  
Intermediate Step ◽  
Infinite Matrix ◽  
Natural Basis ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
Genus 2 ◽  
Lower Triangular ◽  
Invertible Elements

In this paper we present two new bases, [Formula: see text] and [Formula: see text], for the Kauffman bracket skein module of the handlebody of genus 2 [Formula: see text], KBSM[Formula: see text]. We start from the well-known Przytycki-basis of KBSM[Formula: see text], [Formula: see text], and using the technique of parting we present elements in [Formula: see text] in open braid form. We define an ordering relation on an augmented set [Formula: see text] consisting of monomials of all different “loopings” in [Formula: see text], that contains the sets [Formula: see text], [Formula: see text] and [Formula: see text] as proper subsets. Using the Kauffman bracket skein relation we relate [Formula: see text] to the sets [Formula: see text] and [Formula: see text] via a lower triangular infinite matrix with invertible elements in the diagonal. The basis [Formula: see text] is an intermediate step in order to reach at elements in [Formula: see text] that have no crossings on the level of braids, and in that sense, [Formula: see text] is a more natural basis of KBSM[Formula: see text]. Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds [Formula: see text] that are obtained from [Formula: see text] by surgery, since isotopy moves in [Formula: see text] are naturally described by elements in [Formula: see text].

scholarly journals Kauffman bracket skein module of the connected sum of handlebodies and non-injectivity

2021 ◽  
pp. 2150042
Author(s):  
Hiroaki Karuo
Keyword(s):  
Connected Sum ◽  
Skein Module ◽  
Root Of Unity ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
Sum Formula ◽  
Genus Formula ◽  
Natural Way

For the handlebody [Formula: see text] of genus [Formula: see text], Przytycki studied the (Kauffman bracket) skein module [Formula: see text] of the connected sum [Formula: see text] at [Formula: see text]. One of his results is that, in the case when [Formula: see text] is invertible for any [Formula: see text], a homomorphism [Formula: see text] is an isomorphism, which is induced by a natural way. In this paper, in the case when [Formula: see text], the ground ring is [Formula: see text], and [Formula: see text] is a [Formula: see text]-th root of unity ([Formula: see text]), we show that [Formula: see text] is not injective.

scholarly journals SOME RESULTS ABOUT THE KAUFFMAN BRACKET SKEIN MODULE OF THE TWIST KNOT EXTERIOR

2006 ◽  
Vol 15 (08) ◽  
pp. 1095-1106 ◽  
Author(s):  
RĂZVAN GELCA ◽  
FUMIKAZU NAGASATO
Keyword(s):  
Explicit Form ◽  
Final Section ◽  
Character Variety ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Minimal Relation ◽  
Kauffman Bracket Skein Module

In this paper, we list in explicit form the factoring relations of the Kauffman bracket skein module (KBSM for short) of a twist knot exterior. This is done using curves decorated by characters of irreducible SL(2, ℂ)-representations. In the process, we exhibit a relation which holds in the KBSM of the knot exterior, called the minimal relation. In the final section we prove that, when specializing the variable of the Kauffman bracket at t = -1, the minimal relation becomes the defining polynomial of the SL(2, ℂ)-character variety of the twist knot.

scholarly journals Chebyshev polynomials and the Frohman–Gelca formula

2015 ◽  
Vol 24 (04) ◽  
pp. 1550023 ◽  
Author(s):  
Hoel Queffelec ◽  
Heather M. Russell
Keyword(s):  
Chebyshev Polynomials ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
Sum Formula ◽  
Simple Product

Using Chebyshev polynomials, C. Frohman and R. Gelca introduced a basis of the Kauffman bracket skein module of the torus. This basis is especially useful because the Jones–Kauffman product can be described via a very simple Product-to-Sum formula. Presented in this work is a diagrammatic proof of this formula, which emphasizes and demystifies the role played by Chebyshev polynomials.

KAUFFMAN BRACKET SKEIN MODULE OF A FAMILY OF PRISM MANIFOLDS

2011 ◽  
Vol 20 (01) ◽  
pp. 159-170 ◽  
Author(s):  
MACIEJ MROCZKOWSKI
Keyword(s):  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Reidemeister Moves ◽  
Seifert Manifolds ◽  
Kauffman Bracket Skein Module

Diagrams and Reidemeister moves for links in orientable Seifert manifolds are introduced. Using these diagrams, we compute the Kauffman bracket skein modules of prism manifolds with first homology of order 4. In particular, we show that these modules are free.

scholarly journals A MAYER-VIETORIS THEOREM FOR THE KAUFFMAN BRACKET SKEIN MODULE

1999 ◽  
Vol 08 (06) ◽  
pp. 721-729 ◽  
Author(s):  
WALTER F. LOFARO
Keyword(s):  
Orientable Surface ◽  
Closed Orientable Surface ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

The nth relative Kauffman bracket skein modules are defined and two theorems are given relating them to the Kauffman bracket skein module of a 3-manifold. The first theorem covers the case when the 3-manifold is split along a separating closed orientable surface and the second theorem addresses the case when the surface is nonseparating.

KONSTRUKSI ISLAM DALAM KEKUASAAN NEGARA

1970 ◽  
Vol 6 (2) ◽  
Author(s):  
Nurul Aini Musyarofah
Keyword(s):  
Islamic Law ◽  
Social Condition ◽  
The Other ◽  
Good Society ◽  
Ideal Form ◽  
Other Hand ◽  
Islamic Politics ◽  
The Relationship ◽  
The Ideal

The relationship between Islam and state raises a controversy that includes two main groups;formalists and substantialists. Both of them intend to achieve a good social condition which is inaccordance with Islamic politics. The ideal form of good society to be achieved is principallydescribed in the main source of Islamic law, Al Qur’an and As Sunnah, as follows. A form of goodsociety should supprot equality and justice, egalitarianism, and democracy in its social community.The next problem is what the needed methods and instruments to achieve the ideal Islamic politicsare. In this case, the debate on the formalization and substance of Islamic teaching is related to therunning formal political institution.Each group claims itself to be the most representative to the ideal Islam that often leads to anescalating conflict. On the other hand thr arguments of both groups does not reach the wholeMuslims. As a result, the discourse of Islam and state seems to be elitist and political. As a result,Both groups suspect each other each other and try to utilize the controversy on the relationshipbetween Islam and state to get their own benefit which has no relation with the actualization ofIslamic teaching.

Polynomial invariants for virtual marked graphs and virtual surface-link theory I

2017 ◽  
Vol 26 (12) ◽  
pp. 1750081
Author(s):  
Sang Youl Lee
Keyword(s):  
Natural Extension ◽  
Polynomial Invariants ◽  
Kauffman Bracket ◽  
Virtual Links ◽  
Link Theory ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

Esteem and self-esteem as an interweaving polarity. Max Weber´s analysis from the Protestant ethic to the ideal-type of politician

Human Affairs ◽  
2020 ◽  
Vol 30 (3) ◽  
pp. 353-364
Author(s):  
Cristiana Senigaglia
Keyword(s):  
Social Life ◽  
Max Weber ◽  
Modern Society ◽  
Self Esteem ◽  
Protestant Ethic ◽  
Social Reputation ◽  
Conduct Of Life ◽  
The Social ◽  
The Relationship ◽  
The Ideal

AbstractAlthough Max Weber does not specifically analyze the topic of esteem, his investigation of the Protestant ethic offers interesting insights into it. The change in mentality it engendered essentially contributed to enhancing the meaning and importance of esteem in modern society. In his analysis, Weber ascertains that esteem was fundamental to being accepted and integrated into the social life of congregations. Nevertheless, he also highlights that esteem was supported by a form of self-esteem which was not simply derived from a good social reputation, but also achieved through a deep and continual self-analysis as well as a strict discipline in the ethical conduct of life. The present analysis reconstructs the different aspects of the relationship between social and self-esteem and analyzes the consequences of that relationship by focusing on the exemplary case of the politician’s personality and ethic.

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