Representations of the Kauffman bracket skein algebra of the punctured torus

2014 ◽  
Vol 225 (1) ◽  
pp. 45-55 ◽  
Author(s):  
Jea-Pil Cho ◽  
Răzvan Gelca
Keyword(s):  
Kauffman Bracket ◽  
Punctured Torus

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.

scholarly journals Unicity for representations of the Kauffman bracket skein algebra

2018 ◽  
Vol 215 (2) ◽  
pp. 609-650 ◽  
Author(s):  
Charles Frohman ◽  
Joanna Kania-Bartoszynska ◽  
Thang Lê
Keyword(s):  
Kauffman Bracket

scholarly journals A new identity for $\textrm{SL}(2,\mathbf{C})$-characters of the once punctured torus group

2015 ◽  
Vol 22 (2) ◽  
pp. 485-499 ◽  
Author(s):  
Hengnan Hu ◽  
Ser Peow Tan ◽  
Ying Zhang
Keyword(s):  
Punctured Torus

scholarly journals A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

2019 ◽  
Vol 28 (14) ◽  
pp. 1950083 ◽  
Author(s):  
Takeyoshi Kogiso ◽  
Michihisa Wakui
Keyword(s):  
Kauffman Bracket ◽  
Rational Tangles ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Zigzag Type

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1001-1023 ◽  
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Closed Form ◽  
Explicit Expression ◽  
Computer Algebra ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Closed Form Formula

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

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