scholarly journals Classifying and applying rational knots and rational tangles

2002 ◽  
pp. 223-259 ◽  
Author(s):  
Louis H. Kauffman ◽  
Sofia Lambropoulou
Keyword(s):  
Rational Tangles ◽  
Rational Knots

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.

scholarly journals -DEFORMED RATIONALS AND -CONTINUED FRACTIONS

2020 ◽  
Vol 8 ◽  
Author(s):  
SOPHIE MORIER-GENOUD ◽  
VALENTIN OVSIENKO
Keyword(s):  
Continued Fractions ◽  
Jones Polynomial ◽  
Total Positivity ◽  
Recursive Formula ◽  
Rational Numbers ◽  
Binomial Coefficients ◽  
Indecomposable Representation ◽  
Pascal Triangle ◽  
Rational Knots ◽  
Gaussian Binomial Coefficients

We introduce a notion of $q$ -deformed rational numbers and $q$ -deformed continued fractions. A $q$ -deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$ -deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$ -rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$ -deformation of the Farey graph, matrix presentations and $q$ -continuants are given, as well as a relation to the Jones polynomial of rational knots.

scholarly journals Rational Tangles

1997 ◽  
Vol 18 (3) ◽  
pp. 300-332 ◽  
Author(s):  
Jay R. Goldman ◽  
Louis H. Kauffman
Keyword(s):  
Rational Tangles

Tangle equations involving Montesinos links

2021 ◽  
Author(s):  
Hyeyoung Moon ◽  
Isabel K. Darcy
Keyword(s):  
Rational Tangles

Generalized Montesinos tangles are classified, and the system of unoriented tangle equations [Formula: see text] and [Formula: see text] is solved for a generalized Montesinos tangle [Formula: see text] where [Formula: see text] and [Formula: see text] are rational tangles and [Formula: see text] and [Formula: see text] are Montesinos knots/links.

scholarly journals AN OBSTRUCTION TO EMBEDDING 4-TANGLES IN LINKS

1999 ◽  
Vol 08 (03) ◽  
pp. 321-352 ◽  
Author(s):  
DAVID A. KREBES
Keyword(s):  
Continued Fraction ◽  
Unit Cube ◽  
Greatest Common Divisor ◽  
Kauffman Bracket ◽  
Rational Tangles ◽  
Fourth Root

We consider the ways in which a 4-tangle T inside a unit cube can be extended outside the cube into a knot or link L. We present two links n(T) and d(T) such that the greatest common divisor of the determinants of these two links always divides the determinant of the link L. In order to prove this result we give a two-integer invariant of 4-tangles. Calculations are facilitated by viewing the determinant as the Kauffman bracket at a fourth root of -1, which sets the loop factor to zero. For rational tangles, our invariant coincides with the value of the associated continued fraction.

scholarly journals Virtual rational tangles

2020 ◽  
Vol 29 (06) ◽  
pp. 2050034
Author(s):  
Blake Mellor ◽  
Sean Nevin
Keyword(s):  
Recursive Formula ◽  
Rational Tangles ◽  
Bracket Polynomial ◽  
Complex Valued

We use Kauffman’s bracket polynomial to define a complex-valued invariant of virtual rational tangles that generalizes the well-known fraction invariant for classical rational tangles. We provide a recursive formula for computing the invariant, and use it to compute several examples.

FIBERED TORTI-RATIONAL KNOTS

2010 ◽  
Vol 19 (10) ◽  
pp. 1291-1353 ◽  
Author(s):  
MIKAMI HIRASAWA ◽  
KUNIO MURASUGI
Keyword(s):  
Arbitrary Number ◽  
Continued Fraction ◽  
Alexander Polynomials ◽  
Minimal Genus ◽  
Seifert Surfaces ◽  
Tunnel Number ◽  
Rational Knots ◽  
Geometric Techniques ◽  
Continued Fraction Expansions ◽  
Satellite Knots

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

PALINDROME PRESENTATIONS OF RATIONAL KNOTS

2009 ◽  
Vol 18 (03) ◽  
pp. 343-361 ◽  
Author(s):  
ALBERTO CAVICCHIOLI ◽  
DUŠAN REPOVŠ ◽  
FULVIA SPAGGIARI
Keyword(s):  
Alexander Polynomial ◽  
Character Variety ◽  
Simple Alternative ◽  
Rational Knots

We give explicit palindrome presentations of the groups of rational knots, i.e. presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2, ℂ) character variety of twist knots.

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