The bracket polynomial

2018 ◽  
pp. 79-93
Author(s):  
Heather A. Dye
Keyword(s):  
Bracket Polynomial

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 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.

Three approaches to a bracket polynomial for singular links

2017 ◽  
Vol 10 (2) ◽  
pp. 197-218
Author(s):  
Carmen Caprau ◽  
Alex Chichester ◽  
Patrick Chu
Keyword(s):  
Bracket Polynomial

scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

An enhanced bracket polynomial for knotoids

2019 ◽  
Vol 28 (10) ◽  
pp. 1950061
Author(s):  
Yasuyuki Miyazawa
Keyword(s):  
Virtual Link ◽  
Polynomial Invariant ◽  
Link Invariant ◽  
Pole Diagram ◽  
Bracket Polynomial

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

scholarly journals q-DEFORMED SPIN NETWORKS, KNOT POLYNOMIALS AND ANYONIC TOPOLOGICAL QUANTUM COMPUTATION

2007 ◽  
Vol 16 (03) ◽  
pp. 267-332 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Algorithms ◽  
Jones Polynomial ◽  
Braid Groups ◽  
Spin Networks ◽  
Spin Network ◽  
Topological Quantum ◽  
Jones Polynomials ◽  
Topological Quantum Computation ◽  
Bracket Polynomial ◽  
Colored Jones Polynomials

We review the q-deformed spin network approach to Topological Quantum Field Theory and apply these methods to produce unitary representations of the braid groups that are dense in the unitary groups. Our methods are rooted in the bracket state sum model for the Jones polynomial. We give our results for a large class of representations based on values for the bracket polynomial that are roots of unity. We make a separate and self-contained study of the quantum universal Fibonacci model in this framework. We apply our results to give quantum algorithms for the computation of the colored Jones polynomials for knots and links, and the Witten–Reshetikhin–Turaev invariant of three manifolds.

scholarly journals AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (10) ◽  
pp. 1369-1422 ◽  
Author(s):  
LOUIS H. KAUFFMAN
Keyword(s):  
Finite Number ◽  
Crossing Number ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Polynomial Coefficients ◽  
Bracket Polynomial ◽  
Knots And Links

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

2011 ◽  
Vol 20 (03) ◽  
pp. 435-462 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Link Diagram ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Marked Graph ◽  
Bracket Polynomial

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

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