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.

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.

The Kauffman bracket skein module ofS 1×S 2

1995 ◽  
Vol 220 (1) ◽  
pp. 65-73 ◽  
Author(s):  
Jim Hoste ◽  
Józef H. Przytycki
Keyword(s):  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

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 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.

scholarly journals The Yang-Mills measure in the Kauffman bracket skein module

2003 ◽  
Vol 78 (1) ◽  
pp. 1-17 ◽  
Author(s):  
D Bullock ◽  
Joanna Kania-Bartoszynska ◽  
Charles Frohman
Keyword(s):  
Yang Mills ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module

scholarly journals Categorification of the Kauffman bracket skein module ofI–bundles over surfaces

2004 ◽  
Vol 4 (2) ◽  
pp. 1177-1210 ◽  
Author(s):  
Marta M Asaeda ◽  
Jozef H Przytycki ◽  
Adam S Sikora
Keyword(s):  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module
Sign in / Sign up

Export Citation Format

Share Document