Related papers: From tangle fractions to DNA

The connection between continued fractions and orthogonality which is familiar for $J$-fractions and $T$-fractions is extended to what we call $R$-fractions of type I and II. These continued fractions are associated with recurrence…

This paper is base on talks which I gave in May, 2010 at Workshop in Trieste (ICTP). In the first part we present an introduction to knots and knot theory from an historical perspective, starting from Summerian knots and ending on Fox…

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples…

We study two systems of tangle equations that arise when modeling the action of the Integrase family of proteins on DNA. These two systems--direct and inverted repeats--correspond to two different possibilities for the initial DNA sequence.…

In this paper, We introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced…

We examine the conformations of a model for a short segment of closed DNA. The molecule is represented as a cylindrically symmetric elastic rod with a constraint corresponding to a specification of the linking number. We obtain analytic…

This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in…

We define new invariants of knots by means of quandle colorings and longitudinal information. These invariants can be applied to a tangle embedding problem and recognizing non-classical virtual knots.

Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper…

There is a map, defined and studied by Jones, from Thompson's group $F$ to knots. Jones proved that every knot is in the image of this map -- that is, that every knot can be seen as the "knot closure" of a Thompson group element. We…

Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is…

New invariants for 2-dimensional cell complexes are defined, which can be interpreted as curvature bounds. These invariants are proved to be rational and computable in a companion article. This document is a survey that collects theorems…

Tangles were originally introduced as a concept to formalize regions of high connectivity in graphs. In recent years, they have also been discovered as a link between structural graph theory and data science: when interpreting similarity in…

In this paper, we study a geometric/topological measure of knots and links called the nullification number. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, one can intuitively regard…

The homology and cohomology of quandles and racks are used in knot theory: given a finite quandle and a cocycle, we can construct a knot invariant. This is a quick introductory survey to the invariants of knots derived from quandles and…

We show that the problem of constructing a real rational knot of a reasonably low degree can be reduced to an algebraic problem involving the pure braid group: expressing an associated element of the pure braid group in terms of the…

Integrase proteins acting on circular double-stranded DNA often change its topology by transforming unknotted circles into torus knots and links. Two systems of tangle equations--corresponding to the two initial DNA sequences--arise when…

Using the recently developed mathematical theory of tangles, we re-assess the mathematical foundations for applications of the five factor model in personality tests by a new, mathematically rigorous, quantitative method. Our findings…

The fundamental quandle is a powerful invariant of knots and links, but it is difficult to describe in detail. It is often useful to look at quotients of the quandle, especially finite quotients. One natural quotient introduced by Joyce is…

We introduce and study knots and links in 2-dimensional complexes. In particular, we define linking numbers for oriented two-component links in 2-complexes and a Kauffman-type bracket polynomial for links in 2-complexes. We also discuss…