-DEFORMED RATIONALS AND -CONTINUED FRACTIONS

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.

The character variety of some classes of rational knots

We propose a method to determine the character variety of a class [Formula: see text] of rational knots, which includes the twist knots. The defining polynomials depend only on the variables [Formula: see text] and [Formula: see text]. This answers for these classes of knots a question posed in [H. M. Hilden, M. T. Lozano and J. M. Montesinos–Amilibia, On the character variety of group representations of a 2-bridge link [Formula: see text] into [Formula: see text], Bol. Soc. Mat. Mexicana 37(2) (1992) 241–262], and allows us to give an easy geometrical description of the considered character variety. Our results are obtained by using special presentations of the knot groups whose relators are palindromes.

Classification of real rational knots of low degree in the 3-sphere

In this paper, we classify, up to rigid isotopy, real rational knots of degrees less than or equal to [Formula: see text] in a real quadric homeomorphic to the 3-sphere. We also study their connections with rigid isotopy classes of real rational knots of low degree in [Formula: see text] and classify real rational curves of degree 6 in the 3-sphere with exactly one ordinary double point.

Computation of Gordian distances and H2-Gordian distances of knots

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.

THE TENEVA GAME

For each prime p > 7 we obtain the expression for an upper bound on the minimum number of colors needed to non-trivially color T(2, p), the torus knot of type (2, p), modulo p. This expression is t + 2l -1 where t and l are extracted from the prime p. It is obtained from iterating the so-called Teneva transformations which we introduced in a previous article. With the aid of our estimate we show that the ratio "number of colors needed vs. number of colors available" tends to decrease with increasing modulus p. For instance as of prime 331, the number of colors needed is already one tenth of the number of colors available. Furthermore, we prove that 5 is the minimum number of colors needed to non-trivially color T(2, 11) modulo 11. Finally, as a preview of our future work, we prove that 5 is the minimum number of colors modulo 11 for two rational knots with determinant 11.

REAL ALGEBRAIC KNOTS OF LOW DEGREE

In this paper, we study rational real algebraic knots in ℝP3. We show that two real rational algebraic knots of degree ≤ 5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any smooth irreducible knot which admits a plane projection with less than or equal to four crossings has a rational parametrization of degree ≤6. Furthermore an explicit construction of rational knots of a given degree with arbitrary encomplexed writhe (subject to natural restrictions) is presented.