From tangle fractions to DNA

Non-hyperbolic solutions to tangle equations involving composite links

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations $N(O+X_1)=b_1$ and $N(O+X_2)=b_2 \# b_3$, where $X_1$ and $X_2$ are rational tangles, and $b_i$ is a 2-bridge link, for $i=1,2,3$, with $b_2$ and $b_3$ nontrivial. We solve this system of equations under the assumption $\widetilde{O}$, the double branched cover of $O$, is not hyperbolic, i.e.$O$ is not $π$-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption $O$ is an algebraic tangle.

Growth functions of Coxeter groups, Poincare series of singularities and $2$-tangle $q$-fractions

Growth functions of Coxeter groups and the Poincare series of Kleinian and Fuchsian singularities are $2$-tangle $q$-fractions.

缠结方程、Jones猜想、缠结补空间中曲面的斜率以及q-变形有理数

Tangle Equations, the Jones conjecture, slopes of surfaces in tangle complements, and q-deformed rationals

本文研究2-缠结方程组的解的唯一性，证明带标架有理解至多一个，并指出Jones解结猜想蕴含一般唯一性；同时引入Kauffman括号比并猜想其与曲面斜率及q-有理数的关系，还关联了两个著名猜想。

Hard Unknots and Collapsing Tangles

This paper gives infinitely many examples of unknot diagrams that are hard, in the sense that the diagrams need to be made more complicated by Reidemeister moves before they can be simplified. In order to construct these diagrams, we prove theorems characterizing when the numerator of the sum of two rational tangles is an unknot. The key theorem shows that the numerator of the sum of two rational tangles [P/Q] and [R/S] is unknotted if and only if PS + QR has absolute value equal to 1. The paper uses these results in studying processive DNA recombination, finding minimal size unknot diagrams, …

Virtual Rational Tangles

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.

扭绳机、树与结

Taffy, Trees, and Tangles

本文探讨了扭绳机、分数树和有理结之间的关系，提出了扭绳类比并建立了有理结与扭绳操作的几何联系。

An Obstruction to Embedding 4-Tangles in Links

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.

无缠结排列与随机覆盖的 Putman-Wieland 性质

Tangle free permutations and the Putman-Wieland property of Random covers

本文研究曲面随机覆盖，证明高次覆盖中满足 Putman-Wieland 性质的比例趋于 1，且随机覆盖的单值群渐近几乎必然无缠结。

On Rational Knots and Links in the Solid Torus

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.

Unexpected connections between Burnside Groups and Knot Theory

In Classical Knot Theory and in the new Theory of Quantum Invariants substantial effort was directed toward the search for unknotting moves on links. We solve, in this note, several classical problems concerning unknotting moves. Our approach uses a new concept, Burnside groups of links, which establishes unexpected relationship between Knot Theory and Group Theory. Our method has the potential to be used in computational biology in the analysis of DNA via tangle embedding theory, as developed by D.W.Sumners.

绳索、分数与模空间

Ropes, fractions, and moduli spaces

本文阐述康威绳结戏法，揭示其数学原理，并建立三股辫子与椭圆曲线的联系，从而追溯至19世纪魏尔斯特拉斯、阿贝尔和雅可比的工作。

Algebra of Families of Alternating Knots and Links

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and the minimal number of conways. The continued fraction associated to rational knots is represented by gaussian brackets and products of 2-dimensional matrices. This gives birth to an algebra of rational knots and tangles which is easily generalized to alternating knots. A collection of 65 families of prime alternating knots with one to six conways is found. E…

On the Classification of Rational Knots

This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the calculus of continued fractions. We characterize the class of strongly invertible rational links. Rational links are of fundamental importance in the study of DNA recombination.