Constructing real rational knots by gluing

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.

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Generating functions, Fibonacci numbers and rational knots

Journal of Algebra ◽

10.1016/j.jalgebra.2006.11.031 ◽

2007 ◽

Vol 310 (2) ◽

pp. 491-525 ◽

Cited By ~ 3

Author(s):

A. Stoimenow

Keyword(s):

Generating Functions ◽

Fibonacci Numbers ◽

Rational Knots

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Kauffman Bracket on Rational Tangles and Rational Knots

Geometry Integrability and Quantization ◽

10.7546/giq-19-2018-75-90 ◽

2018 ◽

Vol 19 ◽

pp. 75-90

Author(s):

Khaled Bataineh ◽

Keyword(s):

Kauffman Bracket ◽

Rational Tangles ◽

Rational Knots

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FIBERED TORTI-RATIONAL KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216510008431 ◽

2010 ◽

Vol 19 (10) ◽

pp. 1291-1353 ◽

Cited By ~ 1

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.

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PALINDROME PRESENTATIONS OF RATIONAL KNOTS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216509006963 ◽

2009 ◽

Vol 18 (03) ◽

pp. 343-361 ◽

Cited By ~ 1

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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Computation of Gordian distances and H2-Gordian distances of knots

Yugoslav journal of operations research ◽

10.2298/yjor131210044z ◽

2015 ◽

Vol 25 (1) ◽

pp. 133-152 ◽

Cited By ~ 4

Author(s):

Ana Zekovic

Keyword(s):

Knot Theory ◽

Np Hard ◽

Minimal Representations ◽

Rational Knots ◽

Knot Diagrams

One of the most complicated problems in Knot theory is to compute unknotting number. Hass, Lagarias and Pippenger proved that the unknotting problem is NP hard. In this paper we discuss the question of computing unknotting number from minimal knot diagrams, Bernhard-Jablan Conjecture, unknown knot distances between non-rational knots, and searching for minimal distances by using a graph with weighted edges, which represents knot distances. Since topoizomerazes are enzymes involved in changing crossing of DNA, knot distances can be used to study topoizomerazes actions. In the existing tables of knot smoothing, knots with smoothing number 1 are computed by Abe and Kanenobu [27] for knots with at most n = 9 crossings, and smoothing knot distances are computed by Kanenobu [26] for knots with at most n = 7 crossings. We compute some undecided knot distances 1 from these papers, and extend the computations by computing knots with smoothing number one with at most n = 11 crossings and smoothing knot distances of knots with at most n = 9 crossings. All computations are done in LinKnot, based on Conway notation and non-minimal representations of knots.

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Determinants of rational knots

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.454 ◽

2009 ◽

Vol Vol. 11 no. 2 (Combinatorics) ◽

Author(s):

Louis H. Kauffman ◽

Pedro Lopes

Keyword(s):

Odd Order ◽

International Audience ◽

Even Order ◽

Rational Knots ◽

Increasing Sequences

Combinatorics International audience We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.

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